This chapter is entirely Track 2, but it neither depends on nor prepares for any other chapter.
The raison d'etre of this chapter
In relativistic astrophysics, as elsewhere in physics, most problems of deep physical interest are too difficult and too complex to be solved exactly. They can be solved only by use of approximation techniques. And of all approximation techniques, the one that has the widest range of application is perturbation theory.
Perturbation calculations are typically long, tedious, and filled with complicated mathematical expressions. Therefore, they are not appropriate for a textbook such as this. Nevertheless, because it is so important that aspiring astrophysicists know how to set up and carry out perturbation calculations in general relativity, the authors have chosen to present one example in detail.
The example chosen is an analysis of adiabatic, radial pulsations of a nonrotating, relativistic star. Two features of this example are noteworthy: (1) it is sufficiently complex to be instructive, but sufficiently simple to be presentable; (2) in the results of the calculation one can discern and quantify the relativistic instability that is so important for modern astrophysics (see Chapter 24).
The calculation presented here is patterned after that of Chandrasekhar (1964a,b), which first revealed the existence of the relativistic instability. For an alternative calculation, based on the concept of "extremal energy," see Appendix B of Harrison, Thorne, Wakano, and Wheeler (1965); and for a calculation based on extremal entropy, see Cocke (1965).
The authors are deeply indebted to Mr. Carlton M. Caves, who found and corrected many errors in the equations of this chapter and of a dozen other chapters.
§26.2. SETTING UP THE PROBLEM
The system to be analyzed is a sphere of perfect fluid, pulsating radially with very small amplitude. To analyze the pulsations one needs (a) the exact equations governing the equilibrium configuration about which the sphere pulsates; (b) a coordinate system for the vibrating sphere that reduces, for zero pulsation amplitude, to the standard Schwarzschild coordinates of the equilibrium sphere; (c) a set of small functions describing the pulsation (radial displacement and velocity, pressure and density perturbations, first-order changes in metric coefficients), in which to linearize; and (d) a set of equations governing the evolution of these "perturbation functions."
a. Equilibrium Configuration
The equations of structure for the equilibrium sphere are those summarized in $23.7\$ 23.7. It will be useful to rewrite them here, with a few changes of notation (use of subscript " oo " to denote "unperturbed configuration"; use of Lambda=-(1)/(2)ln(1-2m//r)\Lambda=-\frac{1}{2} \ln (1-2 m / r) in place of mm in all equations; use of a prime to denote derivatives with respect to rr ):
{:[(26.1a)ds^(2)=-e^(2Phi_(o))dt^(2)+e^(2Lambda_(o)dr^(2)+r^(2)(dtheta^(2)+sin^(2)theta dphi^(2)),)],[(26.1b)Lambda_(o)^(')=(1)/(2r)(1-e^(2Lambda_(o)))+4pi rrho_(o)e^(2Lambda_(o))","],[(26.1c)p_(o)^(')=-(rho_(o)+p_(o))Phi_(o)^(')","],[(26.1d)Phi_(o)^(')=-(1)/(2r)(1-e^(2Lambda_(o)))+4pi rp_(o)e^(2Lambda_(o))]:}\begin{gather*}
d s^{2}=-e^{2 \Phi_{o}} d t^{2}+e^{2 \Lambda_{o} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right),} \tag{26.1a}\\
\Lambda_{o}^{\prime}=\frac{1}{2 r}\left(1-e^{2 \Lambda_{o}}\right)+4 \pi r \rho_{o} e^{2 \Lambda_{o}}, \tag{26.1b}\\
p_{o}^{\prime}=-\left(\rho_{o}+p_{o}\right) \Phi_{o}^{\prime}, \tag{26.1c}\\
\Phi_{o}^{\prime}=-\frac{1}{2 r}\left(1-e^{2 \Lambda_{o}}\right)+4 \pi r p_{o} e^{2 \Lambda_{o}} \tag{26.1d}
\end{gather*}
b. Coordinates for Perturbed Configuration
The gas sphere pulsates in a radial, i.e., spherically symmetric, manner. Consequently, its spacetime geometry must be spherical. In Box 23.3 it is shown that for any spherical spacetime, whether dynamic or static, one can introduce Schwarzschild coordinates with a line element
{:[(26.2)ds^(2)=-e^(2Phi)dt^(2)+e^(2Lambda)dr^(2)+r^(2)(dtheta^(2)+sin^(2)theta dphi^(2))","],[Phi=Phi(t","r)","quad Lambda=Lambda(t","r).]:}\begin{gather*}
d s^{2}=-e^{2 \Phi} d t^{2}+e^{2 \Lambda} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right), \tag{26.2}\\
\Phi=\Phi(t, r), \quad \Lambda=\Lambda(t, r) .
\end{gather*}
Coordinates for perturbed configuration
One uses these coordinates for the pulsating sphere because they reduce to the unperturbed coordinates when the pulsations have zero amplitude.
c. Perturbation Functions
When the pulsations have very small amplitude, the metric coefficients, Phi\Phi and Lambda\Lambda, and the thermodynamic variables p,rhop, \rho, and nn as measured in the fluid's rest frame
Setting up the analysis of stellar pulsations
Equilibrium configuration of star
have very nearly their unperturbed values. Denote by delta Phi,delta Lambda,delta p,delta rho\delta \Phi, \delta \Lambda, \delta p, \delta \rho, and delta n\delta n the perturbations at fixed coordinate locations:
Besides delta Phi,delta Lambda,delta p,delta rho\delta \Phi, \delta \Lambda, \delta p, \delta \rho, and delta n\delta n, one more perturbation function is needed to describe the pulsations: the radial displacement xi\xi of the fluid from its equilibrium position:
A fluid element located at coordinate radius rr in the unperturbed configuration is displaced to coordinate radius r+xi(r,t)r+\xi(r, t) at coordinate time tt in the vibrating configuration.
To make the analysis of the pulsations tractable, all equations will be linearized in the perturbation functions xi,delta Phi,delta Lambda,delta p,delta rho\xi, \delta \Phi, \delta \Lambda, \delta p, \delta \rho, and delta n\delta n.
d. Equations of Evolution
The evolution of the perturbation functions with time will be governed by the Einstein field equations, the local law of conservation of energy-momentum grad*T=\boldsymbol{\nabla} \cdot \boldsymbol{T}= 0 , and the laws of thermodynamics-all appropriately linearized. The analysis from here on is nothing but a reduction of those equations to "manageable form." Of course, the reduction will proceed most efficiently if one knows in advance what form one seeks. The goal in this calculation and in most similar calculations is simple: (1) obtain a set of dynamic equations for the true dynamic degrees of freedom (only the fluid displacement xi\xi in this case; the fluid displacement and the amplitudes of the gravitational waves in a nonspherical case, where waves are possible); and (2) obtain a set of initial-value equations expressing the remaining perturbation functions ( delta Phi,delta Lambda,delta p,delta rho\delta \Phi, \delta \Lambda, \delta p, \delta \rho, and delta n\delta n in this case) in terms of the dynamic degrees of freedom ( xi\xi ).
§26.3. EULERIAN VERSUS LAGRANGIAN PERTURBATIONS
Before deriving the dynamic and initial-value equations, it is useful to introduce a new concept: the "Lagrangian perturbation" in a thermodynamic variable. The perturbations delta p,delta rho\delta p, \delta \rho, and delta n\delta n of equations (26.3) are Eulerian perturbations in p,rhop, \rho, and nn; i.e., they are changes measured by an observer who sits forever at a fixed point (t,r,theta,phi)(t, r, \theta, \phi) in the coordinate grid. By contrast, the Lagrangian perturbationsdenoted Delta p,Delta rho\Delta p, \Delta \rho, and Delta n\Delta n-are changes measured by an observer who moves with
Eulerian perturbations defined
Lagrangian perturbations defined
How to derive equations governing the perturbation functions
the fluid; i.e., by an observer who would sit at radius rr in the unperturbed configuration, but sits at r+xi(t,r)r+\xi(t, r) in the perturbed configuration:
Relation between Eulerian and Lagrangian perturbations
§26.4. INITIAL-VALUE EQUATIONS
a. Baryon Conservation
The law of baryon conservation, grad*(nu)=0\boldsymbol{\nabla} \cdot(n \boldsymbol{u})=0 ( $22.2\$ 22.2 ), governs the evolution of perturbations Delta n\Delta n and delta n\delta n in baryon number. By applying the chain rule to the divergence and using the relation u*grad n=grad_(u)n=dn//d tau\boldsymbol{u} \cdot \boldsymbol{\nabla} n=\boldsymbol{\nabla}_{\boldsymbol{u}} n=d n / d \tau, one can rewrite the conservation law as
{:[dn//d tau=-n(grad*u).],[^(uarr)" [derivative of "n" along fluid world line] "]:}\begin{aligned}
& d n / d \tau=-n(\boldsymbol{\nabla} \cdot \boldsymbol{u}) . \\
& \stackrel{\uparrow}{ } \text { [derivative of } n \text { along fluid world line] }
\end{aligned}
In terms of Delta n\Delta n, the perturbation measured by an observer moving with the fluid, this equation can be rewritten as
To reduce this equation further, one needs an expression for the fluid's 4 -velocity. It is readily derived from
{:[(u^(r))/(u^(t))=((dr//d tau)/(dt//d tau))=((dr)/(dt))_("along world line ")=(del xi)/(del t)-=xi^(˙)","],[(u^(t))^(2)e^(2varphi)-(u^(r))^(2)e^(2Lambda)=1.]:}\begin{gathered}
\frac{u^{r}}{u^{t}}=\left(\frac{d r / d \tau}{d t / d \tau}\right)=\left(\frac{d r}{d t}\right)_{\text {along world line }}=\frac{\partial \xi}{\partial t} \equiv \dot{\xi}, \\
\left(u^{t}\right)^{2} e^{2 \varphi}-\left(u^{r}\right)^{2} e^{2 \Lambda}=1 .
\end{gathered}
The result to first order in xi,delta Lambda\xi, \delta \Lambda, and delta Phi\delta \Phi is
Derivation of initial value equations:
(1) for baryon perturbations Delta n\Delta n and delta n\delta n
This is the initial value equation for Delta n\Delta n in terms of the dynamic variable xi\xi. The initial-value equation for delta n\delta n, which will not be needed later, one obtains by combining with equation (26.4c).
b. Adiabaticity
(2) for pressure perturbations Delta p\Delta p and delta p\delta p
For adiabatic vibrations (negligible heat transfer between neighboring fluid elements), the Lagrangian changes in number density and pressure are related by
Combining this adiabatic relation with equation (26.7) for Delta n\Delta n, and equation (26.4a) for delta p\delta p in terms of Delta p\Delta p, one obtains the following initial-value equation for delta p\delta p :
The local law of energy conservation [first law of thermodynamics; u*(grad*T)=0\boldsymbol{u} \cdot(\boldsymbol{\nabla} \cdot \boldsymbol{T})=0; see §§22.2\S \S 22.2 and 22.3] says that
Rewritten in terms of Lagrangian perturbations (recall: d//d taud / d \tau is a time derivative as measured by an observer moving with the fluid), this reads
which has as its time integral (first-order analysis!)
{:(26.10)Delta rho=(rho_(o)+p_(o))/(n_(o))Delta n.:}\begin{equation*}
\Delta \rho=\frac{\rho_{o}+p_{o}}{n_{o}} \Delta n . \tag{26.10}
\end{equation*}
(The constant of integration is zero, because, when Delta n=0,Delta rho\Delta n=0, \Delta \rho must also vanish.) Combining this with equation (26.7) for Delta n\Delta n and equation (26.4b) for delta rho\delta \rho in terms of Delta rho\Delta \rho, one obtains the following initial-value equation for delta rho\delta \rho :
Two of the Einstein field equations, when linearized, reduce to initial-value equations for the metric perturbations delta Lambda\delta \Lambda and delta Phi\delta \Phi. The equations needed, expressed in an orthonormal frame
are G_( hat(r) hat(t))=8piT_( hat(r) hat(t))G_{\hat{r} \hat{t}}=8 \pi T_{\hat{r} \hat{t}}, and G_( hat(r) hat(r))=8piT_( hat(r) hat(r))G_{\hat{r} \hat{r}}=8 \pi T_{\hat{r} \hat{r}}. The components of the Einstein tensor in this orthonormal frame were evaluated in exercise 14.16:
The components of the stress-energy tensor, T_( hat(alpha) hat(beta))=(rho+p)u_( hat(alpha))u_( hat(beta))+peta_(alpha beta)T_{\hat{\alpha} \hat{\beta}}=(\rho+p) u_{\hat{\alpha}} u_{\hat{\beta}}+p \eta_{\alpha \beta}, as calculated using the 4 -velocity (26.6) [transformed into the form u_( hat(o))=-1,u_( hat(r))=xi^(˙)e^(Lambda_(o)-phi_(o))u_{\hat{o}}=-1, u_{\hat{r}}=\dot{\xi} e^{\Lambda_{o}-\phi_{o}} ] and using expressions (26.3a) for rho\rho and pp, reduce to
Consequently, the field equation G_( hat(r) hat(t))=8piT_( hat(r) hat(t))G_{\hat{r} \hat{t}}=8 \pi T_{\hat{r} \hat{t}}-after integration with respect to time and choice of the constant of integration, so that delta Lambda=0\delta \Lambda=0 when xi=0\xi=0-reduces to
This is the initial-value equation for delta Lambda\delta \Lambda. The field equation G_( hat(r) hat(r))=8piT_( hat(r) hat(r))G_{\hat{r} \hat{r}}=8 \pi T_{\hat{r} \hat{r}}, after using (26.15) to remove delta Lambda\delta \Lambda and (26.9) to remove delta p\delta p, and (26.1c) to remove Phi_(o)^(')\Phi_{o}{ }^{\prime}, reduces to
This is the initial-value equation for delta Phi\delta \Phi.
§26.5. DYNAMIC EQUATION AND BOUNDARY CONDITIONS
The dynamic evolution of the fluid displacement xi(t,r)\xi(t, r) is governed by the Euler equation (22.13):
(rho+p)xx(4"-acceleration ")=-(" projection of "grad p" orthogonal to "u)(\rho+p) \times(4 \text {-acceleration })=-(\text { projection of } \boldsymbol{\nabla} p \text { orthogonal to } \boldsymbol{u})
The 4-acceleration a=grad_(u)u\boldsymbol{a}=\boldsymbol{\nabla}_{\boldsymbol{u}} \boldsymbol{u} corresponding to the 4-velocity (26.6) in the metric (26.2) has as its only non-zero, linearized, covariant component:
(4) for metric perturbations delta Lambda\delta \Lambda and delta Phi\delta \Phi
[The component a_(t)a_{t} is trivial in the sense that it leads to an Euler equation that duplicates (26.1c).] Combining this with rho+p=rho_(o)+p_(o)+delta rho+delta p\rho+p=\rho_{o}+p_{o}+\delta \rho+\delta p, with the radial component p_(o)^(')+deltap^(')p_{o}{ }^{\prime}+\delta p^{\prime} for the projection of grad p\boldsymbol{\nabla} p, and with the zero-order equation of hydrostatic equilibrium (26.1c), one obtains for the Euler equation
This equation of motion is put into its most useful form by using the initial-value equations (26.9), (26.11), and (26.16) to reexpress delta p,delta rho\delta p, \delta \rho, and deltaPhi^(')\delta \Phi^{\prime} in terms of xi\xi, and by then manipulating terms extensively with the aid of the zero-order equations of structure (26.1). The result is
{:(26.19)Wzeta^(¨)=(Pzeta^('))^(')+Q zeta",":}\begin{equation*}
W \ddot{\zeta}=\left(P \zeta^{\prime}\right)^{\prime}+Q \zeta, \tag{26.19}
\end{equation*}
where zeta\zeta is a "renormalized displacement function," and W,P,QW, P, Q are functions of radius determined by the structure of the equilibrium star:
Equation (26.19) is the dynamic equation governing the stellar pulsations. [This equation could be written in other forms; for instance, it could be multiplied by W^(-1)W^{-1} or any other non-zero factor, and terms could be regrouped. The form given in equation (26.19) is preferred because it leads to a self-adjoint eigenvalue problem for the oscillation frequencies, as indicated in Box 26.1.]
Not all solutions of the dynamic equation are acceptable. To be physically acceptable, the displacement function must produce noninfinite density and pressure perturbations ( delta rho\delta \rho and delta p\delta p ) at the center of the sphere, which means
{:(26.22a)(xi//r)" finite or zero in limit as "r longrightarrow0:}\begin{equation*}
(\xi / r) \text { finite or zero in limit as } r \longrightarrow 0 \tag{26.22a}
\end{equation*}
[see (26.9) and (26.11)]; also, it must leave the pressure equal to zero at the star's surface, which means
If an initial displacement of the fluid, xi(t=0,r)\xi(t=0, r), is specified subject to the boundary conditions (26.22), then its subsequent evolution xi(t,r)\xi(t, r) can be calculated by inte-
grating the dynamic equation (26.19); and the form of the pressure, density, and metric perturbations can be calculated from xi(t,r)\xi(t, r) using the initial-value equations (26.9), (26.11), (26.15), and (26.16).
Several important consequences of these results are explored in Boxes 26.1 and 26.2.
(continued on page 699)
Box 26.1 EIGENVALUE PROBLEM AND VARIATIONAL PRINCIPLE FOR NORMAL-MODE PULSATIONS OF A STAR
Assume that the renormalized displacement function (26.20) has a sinusoidal time dependence:
Then the dynamic equation (26.19) and boundary conditions (26.22) reduce to an eigenvalue problem for the angular frequency omega\omega and amplitude zeta(r)\zeta(r) :
{:[(1)(Pzeta^('))^(')+Q zeta+omega^(2)W zeta=0","],[(2a)zeta//r^(3)" finite or zero as "r longrightarrow0","],[(2b)Gamma_(1)p_(0)r^(-2)e^(phi_(0)zeta^(')longrightarrow0" as "r longrightarrow R)]:}\begin{gather*}
\left(P \zeta^{\prime}\right)^{\prime}+Q \zeta+\omega^{2} W \zeta=0, \tag{1}\\
\zeta / r^{3} \text { finite or zero as } r \longrightarrow 0, \tag{2a}\\
\Gamma_{1} p_{0} r^{-2} e^{\phi_{0} \zeta^{\prime} \longrightarrow 0 \text { as } r \longrightarrow R} \tag{2b}
\end{gather*}
Methods for solving this eigenvalue problem are catalogued and discussed by Bardeen, Thorne, and Meltzer (1966). One method (but not the best for numerical calculations) is the variational principle:
{:(3)omega^(2)=" extremal value of "{(int_(0)^(R)(Pzeta^(''2)-Qzeta^(2))dr)/(int_(0)^(R)Wzeta^(2)dr)}:}\begin{equation*}
\omega^{2}=\text { extremal value of }\left\{\frac{\int_{0}^{R}\left(P \zeta^{\prime \prime 2}-Q \zeta^{2}\right) d r}{\int_{0}^{R} W \zeta^{2} d r}\right\} \tag{3}
\end{equation*}
where zeta\zeta is varied over all functions satisfying the boundary conditions (2). [See e.g., §12.3\S 12.3 of Mathews and Walker (1965) for discussion of the equivalence between this variational principle and the original eigenvalue problem.]
The absolute minimum value of expression (3) is the squared frequency of the fundamental mode of pulsation. If it is negative, the star is unstable ( e^(-i omega t)e^{-i \omega t} grows exponentially in time). If it is positive, the star is stable against adiabatic, radial perturbations. Therefore, since the denominator of expression (3) is positive definite,
{:(4)[[" stability against "],[" adiabatic radial "],[" perturbations "]]Longleftrightarrow[int_(0)^(R)(Pzeta^(zeta^(2))-Qzeta^(2))dr > 0" for all functions "zeta" satisfying (2) "]:}\left[\begin{array}{l}
\text { stability against } \tag{4}\\
\text { adiabatic radial } \\
\text { perturbations }
\end{array}\right] \Longleftrightarrow\left[\int_{0}^{R}\left(P \zeta^{\zeta^{2}}-Q \zeta^{2}\right) d r>0 \text { for all functions } \begin{array}{r}
\zeta \text { satisfying (2) }
\end{array}\right]
Box 26.1 (continued)
By numerical solution of the eigenvalue equation (1), the pulsation frequencies have been calculated for a wide variety of models of neutron stars and supermassive stars. Example: The figure gives a plot of pulsation frequency as a function of central density for the lowest four normal modes of the Harrison-Wakano-Wheeler models at the endpoint of stellar evolution. (Make a detailed comparison with Figure 24.2.) These curves are based on calculations by Meltzer and Thorne (1966), with corrections for the fundamental mode of massive white dwarfs by Faulkner and Gribbin (1968).
Box 26.2 THE CRITICAL ADIABATIC INDEX FOR NEARLY NEWTONIAN STARS
A. Fully Newtonian Stars
For a Newtonian star that pulsates sinusoidally, xi=xi(r)e^(-i omega t)\xi=\xi(r) e^{-i \omega t}, the dynamic equation (26.19)(26.19) reduces to
If Gamma_(1)=4//3\Gamma_{1}=4 / 3 throughout the star, the physically acceptable solution [solution satisfying boundary conditions (26.22)] for the fundamental mode of vibration (mode with lowest value of omega^(2)\omega^{2} ) is
Thus, for Gamma_(1)=4//3\Gamma_{1}=4 / 3 the fundamental mode is "neutrally stable" and has a "homologous" displacement function-independent of the star's equation of state or structure.
3. If Gamma_(1)\Gamma_{1} is allowed to differ slightly from 4//34 / 3 in an rr-dependent way, then xi(r)\xi(r) will differ slightly from the homologous form:
xi=epsilon r[1+r"-dependent corrections of magnitude "(Gamma_(1)-4//3)]\xi=\epsilon r\left[1+r \text {-dependent corrections of magnitude }\left(\Gamma_{1}-4 / 3\right)\right]
Consequently, if one uses the homologous expression xi=epsilon r\xi=\epsilon r as a trial function in the variational principle of Box 26.1 , one will obtain omega^(2)\omega^{2} accurate to O[(Gamma_(1)-4//3)^(2)]O\left[\left(\Gamma_{1}-4 / 3\right)^{2}\right]. (Recall: first-order errors in trial function produce second-order errors in value of variational expression.) The Newtonian limit of the variational expression [equation (3) of Box 26.1] becomes, with the homologous choice of trial function,
{:(3)omega^(2)=(3 bar(Gamma)_(1)-4)(int_(0)^(R)3p_(o)r^(2)dr)/(int_(0)^(R)rho_(o)r^(4)dr)+O[(3 bar(Gamma)_(1)-4)^(2)]:}\begin{equation*}
\omega^{2}=\left(3 \bar{\Gamma}_{1}-4\right) \frac{\int_{0}^{R} 3 p_{o} r^{2} d r}{\int_{0}^{R} \rho_{o} r^{4} d r}+O\left[\left(3 \bar{\Gamma}_{1}-4\right)^{2}\right] \tag{3}
\end{equation*}
where bar(Gamma)_(1)\bar{\Gamma}_{1} is the pressure-averaged adiabatic index
{:(4) bar(Gamma)_(1)=(int_(0)^(R)Gamma_(1)p_(o)4pir^(2)dr)/(int_(0)^(R)p_(0)4pir^(2)dr):}\begin{equation*}
\bar{\Gamma}_{1}=\frac{\int_{0}^{R} \Gamma_{1} p_{o} 4 \pi r^{2} d r}{\int_{0}^{R} p_{0} 4 \pi r^{2} d r} \tag{4}
\end{equation*}
Box 26.2 (continued)
By use of the Newtonian virial theorem for the nonpulsating star [equation (39.21b) or exercise 23.7], one can convert equation (3) into the form
{:(5)omega^(2)=(3 bar(Gamma)_(1)-4)|Omega|//I",":}\begin{equation*}
\omega^{2}=\left(3 \bar{\Gamma}_{1}-4\right)|\Omega| / I, \tag{5}
\end{equation*}
where Omega\Omega is the star's self-gravitational energy and I=int(rho_(o)r^(2))4pir^(2)drI=\int\left(\rho_{o} r^{2}\right) 4 \pi r^{2} d r is the trace of the second moment of its mass distribution (see Box 24.2 and exercise 39.6).
B. Nearly Newtonian Stars
When one takes into account first-order relativistic corrections (corrections of order M//RM / R ), but ignores higher-order corrections, one can rewrite the variational expression [equation (3) of Box 26.1] in the form
{:(6)omega^(2)=(int_(0)^(R)p_(o)[Gamma_(1)r^(4)eta^('2)+(3Gamma_(1)-4)(r^(3)eta^(2))^(')](1+Lambda_(o)+3Phi_(o))dr-int_(0)^(R)F_(o)eta^(2)dr)/(int_(0)^(R)rho_(o)r^(4)(1+3Lambda_(o)+Phi_(o)+p_(o)//rho_(o))eta^(2)dr)",":}\begin{equation*}
\omega^{2}=\frac{\int_{0}^{R} p_{o}\left[\Gamma_{1} r^{4} \eta^{\prime 2}+\left(3 \Gamma_{1}-4\right)\left(r^{3} \eta^{2}\right)^{\prime}\right]\left(1+\Lambda_{o}+3 \Phi_{o}\right) d r-\int_{0}^{R} F_{o} \eta^{2} d r}{\int_{0}^{R} \rho_{o} r^{4}\left(1+3 \Lambda_{o}+\Phi_{o}+p_{o} / \rho_{o}\right) \eta^{2} d r}, \tag{6}
\end{equation*}
and m_(o)(r)m_{o}(r) is the equilibrium mass inside radius rr.
2. For a relativistic star with Gamma_(1)-4//3\Gamma_{1}-4 / 3 of order M//RM / R and with M//R≪1M / R \ll 1, the homologous trial function xi=epsilon r\xi=\epsilon r will still be highly accurate. Equally accurate, and easier to work with, will be xi=epsilon re^(phi_(o))~~epsilon r(1+Phi_(o))\xi=\epsilon r e^{\phi_{o}} \approx \epsilon r\left(1+\Phi_{o}\right), which corresponds to eta=epsilon=\eta=\epsilon= constant. Its fractional errors will be of order M//rM / r; and the errors which it produces in omega^(2)\omega^{2} will be of order (M//R)^(2)(M / R)^{2}. By inserting this trial function into the variational principle (6) and keeping only relativistic corrections of order M//RM / R, one obtains
Here bar(Gamma)_(1)\bar{\Gamma}_{1} is the pressure-averaged adiabatic index, and the critical value of the adiabatic index Gamma_(1" crit ")\Gamma_{1 \text { crit }} is
{:(9)Gamma_(1crit)=(4)/(3)+alpha M//R:}\begin{equation*}
\Gamma_{1 \mathrm{crit}}=\frac{4}{3}+\alpha M / R \tag{9}
\end{equation*}
with alpha\alpha a positive constant of order unity given by
{:(10)alpha=(1)/(3)(R)/(M|Omega|)int_(0)^(R)(3rho_(o)(m_(o)^(2))/(r^(2))+4p_(o)(m_(o))/(r))4pir^(2)dr.:}\begin{equation*}
\alpha=\frac{1}{3} \frac{R}{M|\Omega|} \int_{0}^{R}\left(3 \rho_{o} \frac{m_{o}{ }^{2}}{r^{2}}+4 p_{o} \frac{m_{o}}{r}\right) 4 \pi r^{2} d r . \tag{10}
\end{equation*}
Expressions (8) and (9) for the pulsation frequency and the adiabatic index play an important role in the theory of supermassive stars ( $24.4\$ 24.4 ).
3. For alternative derivations of the above result, see Chandrasekhar (1964a,b; 1965c), Fowler (1964, 1965), Wright (1964).
Exercise 26.1. DRAGGING OF INERTIAL FRAMES BY A SLOWLY ROTATING STAR
A fluid sphere rotates very slowly. Analyze its rotation using perturbation theory; keep only effects and terms linear in the angular velocity of rotation. [Hints: (1) Centrifugal forces are second-order in angular velocity. Therefore, to first order the star is undeformed; its density and pressure distributions remain spherical and unperturbed. (2) Show, by symmetry and time-reversal arguments, that one can introduce coordinates in which
{:[(26.23)ds^(2)=-e^(2phi)dt^(2)+e^(2Lambda)dr^(2)+r^(2)[dtheta^(2)+sin^(2)theta dphi^(2)]],[-2(r^(2)sin^(2)theta)omega d phi dt]:}\begin{align*}
d s^{2}= & -e^{2 \phi} d t^{2}+e^{2 \Lambda} d r^{2}+r^{2}\left[d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right] \tag{26.23}\\
& -2\left(r^{2} \sin ^{2} \theta\right) \omega d \phi d t
\end{align*}
where
{:(26.24)Phi=Phi(r)","Lambda=Lambda(r)","" and "omega=omega(r","theta):}\begin{equation*}
\Phi=\Phi(r), \Lambda=\Lambda(r), \text { and } \omega=\omega(r, \theta) \tag{26.24}
\end{equation*}
Show that Phi=Phi_(o)\Phi=\Phi_{o} and Lambda=Lambda_(o)\Lambda=\Lambda_{o} (no perturbations!) to first-order in angular velocity. (3) Adopt the following precise definition of the angular velocity Omega(r,theta)\Omega(r, \theta) :
{:(26.25)Omega-=u^(phi)//u^(t)=(d phi//dt)_("moving with the fluid "):}\begin{equation*}
\Omega \equiv u^{\phi} / u^{t}=(d \phi / d t)_{\text {moving with the fluid }} \tag{26.25}
\end{equation*}
Assuming u^(tau)=u^(theta)=0u^{\tau}=u^{\theta}=0 (i.e., rotation in the phi\phi direction), calculate the 4 -velocity of the fluid. (4) Use the Einstein field equations to derive a differential equation for the metric perturbation omega\omega in terms of the angular velocity Omega\Omega. (5) Solve that differential equation outside the star in terms of elementary functions, and express the solution for omega(r,theta)\omega(r, \theta) in terms of the star's total angular momentum S\boldsymbol{S}, as measured using distant gyroscopes (see Chapter 19).] For the original analyses of this problem and of related topics, see Gurovich (1965), Doroshkevich, Zel'dovich, and Novikov (1965), Hartle and Sharp (1965), Brill and Cohen (1966), Hartle (1967), Krefetz (1967), Cohen and Brill (1968), Cohen (1968).
EXERCISE
THE UNIVERSE
Wherein the reader, flushed with joy at his conquest of the stars, seeks to control the entire universe, and is foiled by an unfathomed mystery: the Initial Singularity.
снартев 27
IDEALIZED COSMOLOGIES
From my point of view one cannot arrive, by way of theory, at any at least somewhat reliable results in the field of cosmology, if one makes no use of the principle of general relativity.
ALBERT EINSTEIN (1949b, p. 684)
§27.1. THE HOMOGENEITY AND ISOTROPY OF THE UNIVERSE
Astronomical observations reveal that the universe is homogeneous and isotropic on scales of ∼10^(8)\sim 10^{8} light years and larger. Taking a "fine-scale" point of view, one sees the agglomeration of matter into stars, galaxies, and clusters of galaxies in regions of size ∼1\sim 1 light year, ∼10^(6)\sim 10^{6} light years, and ∼3xx10^(7)\sim 3 \times 10^{7} light years, respectively. But taking instead a "large-scale" viewpoint, one sees little difference between an elementary volume of the universe of the order of 10^(8)10^{8} light years on a side centered on the Earth and other elementary volumes of the same size located elsewhere.
Cosmology, summarized in its simplest form in Box 27.1, takes the large-scale viewpoint as its first approximation; and as its second approximation, it treats the fine-scale structure as a perturbation on the smooth, large-scale background. This chapter (27) treats in detail the large-scale, homogeneous approximation. Chapter 28 considers such small-scale phenomena as the primordial formation of the elements, and the condensation of galaxies out of the primeval plasma during the expansion of the universe. Chapter 29 discusses observational cosmology.
Evidence for the large-scale homogeneity and isotropy of the universe comes from several sources. (1) There is evidence in the distribution of galaxies on the sky and in the distribution of their apparent magnitudes and redshifts [see, e.g., Hubble (1934b, 1936); Sandage (1972a); Sandage, Tamman, and Hardy (1972); but note the papers claiming "hierarchic" deviations from homogeneity, which Sandage cites and attacks]. (2) There is evidence in the isotropy of the distribution of radio sources on the sky [see, e.g., Holden (1966), and Hughes and Longair (1967)]. (3) There is evidence in the remarkable isotropy of the cosmic microwave radiation [see, e.g., Boughn, Fram, and Partridge (1971)]. For a review of most of the evidence, see Chapter 2 of Peebles (1971).
The universe: fine-scale condensations contrasted with large-scale homogeneity
Evidence for large-scale homogeneity and isotropy
Box 27.1 COSMOLOGY IN BRIEF
Uniform density. Idealize the stars and atoms as scattered like dust through the heavens with an effective average density rho\rho of mass-energy everywhere the same.
Geometry homogeneous and isotropic. Idealize the curvature of space to be everywhere the same.
Closure. Accept the term, "Einstein's geometric theory of gravity" as including not only his field equation G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T}, but also his boundary condition of closure imposed on any solution of this equation.*
A three-sphere satisfies the three requirements of homogeneity, isotropy, and closure, and is the natural generalization of the metric on a circle and a 2 -sphere:
Spheres of selected dimensionality
Visualized as embedded in a Euclidean space of one higher dimension ^("a "){ }^{\text {a }}
Transformation from Cartesian to polar coordinates
Metric on S^(n)S^{n} expressed in terms of these polar coordinates
S^(1)S^{1}
x^(2)+y^(2)=a^(2)x^{2}+y^{2}=a^{2}
{:[x=a cos phi],[y=a sin phi]:}\begin{aligned} & x=a \cos \phi \\ & y=a \sin \phi \end{aligned}
ds^(2)=a^(2)dphi^(2)d s^{2}=a^{2} d \phi^{2}
S^(2)S^{2}
x^(2)+y^(2)+z^(2)=a^(2)x^{2}+y^{2}+z^{2}=a^{2}
{:[x=a sin theta cos phi],[y=a sin theta sin phi],[z=a cos theta]:}\begin{aligned} & x=a \sin \theta \cos \phi \\ & y=a \sin \theta \sin \phi \\ & z=a \cos \theta \end{aligned}
ds^(2)=a^(2)(dtheta^(2)+sin^(2)theta dphi^(2))d s^{2}=a^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)
{:[x=a sin chi sin theta cos phi],[y=a sin chi sin theta sin phi],[z=a sin chi cos theta],[w=a cos theta]:}\begin{aligned} x & =a \sin \chi \sin \theta \cos \phi \\ y & =a \sin \chi \sin \theta \sin \phi \\ z & =a \sin \chi \cos \theta \\ w & =a \cos \theta \end{aligned}
{:[ds^(2)=a^(2)[dchi^(2)+:}],[{:sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(˙)^(2))]]:}\begin{aligned} & d s^{2}=a^{2}\left[d \chi^{2}+\right. \\ & \left.\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \dot{\phi}^{2}\right)\right] \end{aligned}
Spheres of selected dimensionality Visualized as embedded in a Euclidean space of one higher dimension ^("a ") Transformation from Cartesian to polar coordinates Metric on S^(n) expressed in terms of these polar coordinates
S^(1) x^(2)+y^(2)=a^(2) "x=a cos phi
y=a sin phi" ds^(2)=a^(2)dphi^(2)
S^(2) x^(2)+y^(2)+z^(2)=a^(2) "x=a sin theta cos phi
y=a sin theta sin phi
z=a cos theta" ds^(2)=a^(2)(dtheta^(2)+sin^(2)theta dphi^(2))
S^(3) x^(2)+y^(2)+z^(2)+w^(2)=a^(2) "x=a sin chi sin theta cos phi
y=a sin chi sin theta sin phi
z=a sin chi cos theta
w=a cos theta" "ds^(2)=a^(2)[dchi^(2)+:}
{:sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(˙)^(2))]"| Spheres of selected dimensionality | Visualized as embedded in a Euclidean space of one higher dimension ${ }^{\text {a }}$ | Transformation from Cartesian to polar coordinates | Metric on $S^{n}$ expressed in terms of these polar coordinates |
| :---: | :---: | :---: | :---: |
| $S^{1}$ | $x^{2}+y^{2}=a^{2}$ | $\begin{aligned} & x=a \cos \phi \\ & y=a \sin \phi \end{aligned}$ | $d s^{2}=a^{2} d \phi^{2}$ |
| $S^{2}$ | $x^{2}+y^{2}+z^{2}=a^{2}$ | $\begin{aligned} & x=a \sin \theta \cos \phi \\ & y=a \sin \theta \sin \phi \\ & z=a \cos \theta \end{aligned}$ | $d s^{2}=a^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$ |
| $S^{3}$ | $x^{2}+y^{2}+z^{2}+w^{2}=a^{2}$ | $\begin{aligned} x & =a \sin \chi \sin \theta \cos \phi \\ y & =a \sin \chi \sin \theta \sin \phi \\ z & =a \sin \chi \cos \theta \\ w & =a \cos \theta \end{aligned}$ | $\begin{aligned} & d s^{2}=a^{2}\left[d \chi^{2}+\right. \\ & \left.\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \dot{\phi}^{2}\right)\right] \end{aligned}$ |
^("a "){ }^{\text {a }} Excursion off the sphere is physically meaningless and is forbidden. The superfluous dimension is added to help the reason in reasoning, not to help the traveler in traveling. Least of all does it have anything whatsoever to do with time.
The spacetime geometry is described by the metric
{:(1)ds^(2)=-dt^(2)+a^(2)(t)[dchi^(2)+sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{equation*}
d s^{2}=-d t^{2}+a^{2}(t)\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{1}
\end{equation*}
The dynamics of the geometry is known in full when one knows the radius aa as a function of the time tt.
Einstein's field equation (doubled, for convenience), 2G=16 pi T2 \boldsymbol{G}=16 \pi \boldsymbol{T}, has its whole force concentrated in its hat(0) hat(0)\hat{0} \hat{0} (or hat(t) hat(t)\hat{t} \hat{t} ) component,
{:(2)(6)/(a^(2))((da)/(dt))^(2)+(6)/(a^(2))=16 pi rho:}\begin{equation*}
\frac{6}{a^{2}}\left(\frac{d a}{d t}\right)^{2}+\frac{6}{a^{2}}=16 \pi \rho \tag{2}
\end{equation*}
[equation (5a) of Box 14.5]. This component of Einstein's equation is as central as the component grad*E=4pi rho\boldsymbol{\nabla} \cdot \boldsymbol{E}=4 \pi \rho of Maxwell's equations. It is described in the Track-2 Chapter 21 as the "initial-value equation" of geometrodynamics. There the two terms on the left receive separate names: the "second invariant" of the "extrinsic curvature" of a "spacelike slice" through the 4-geometry (tells how rapidly all linear dimensions are being stretched from instant to instant); and the "intrinsic curvature" or threedimensional scalar curvature invariant ^((3))R{ }^{(3)} R of the "spacelike slice" (here a 3 -sphere) at the given instant itself.
The amount of mass-energy in the universe changes from instant to instant in accordance with the work done by pressure during the expansion,
{:(3)d[((" density of ")/(" mass-energy "))xx(" volume ")]=-(" pressure ")d(" volume ").:}\begin{equation*}
d\left[\binom{\text { density of }}{\text { mass-energy }} \times(\text { volume })\right]=-(\text { pressure }) d(\text { volume }) . \tag{3}
\end{equation*}
Today the pressure of radiation is negligible compared to the density of mass-energy, and the righthand side of this equation ("work done") can be neglected. The same was true in the past, one estimates, back to a time when linear dimensions were about a thousand times smaller than they are today. During this "matter-dominated phase" of the expansion of the universe, the product
((" density of ")/(" mass-energy "))xx(" volume ")\binom{\text { density of }}{\text { mass-energy }} \times(\text { volume })
Here the symbol MM can look like mass in the form of matter, and can even be called mass; but one has to recall again (see §19.4) that the concept of total mass-energy of a closed universe has absolutely no well-defined meaning whatsoever, not least because there is no "platform" outside the universe on which to stand to measure its attraction via periods of Keplerian orbits or in any other way. More convenient than MM, because more significant in what follows, is the quantity a_(max)a_{\max } ("radius of universe at phase of maximum expansion") defined by
{:(5)a_(max)=4M//3pi:}\begin{equation*}
a_{\max }=4 M / 3 \pi \tag{5}
\end{equation*}
Box 27.1 (continued)
The decisive component of the Einstein field equation, in the terms of this notation, becomes
The first term in (6) has the qualitative character of "kinetic energy" in an elementary problem in Newtonian mechanics. The second term has the qualitative character of a "potential energy,"
V(a)=-(a_(max))/(a)V(a)=-\frac{a_{\max }}{a}
(see diagram A), resulting from an inverse-square Newtonian force. Pursuing the analogy, one identifies the " -1 " on the righthand side with the total energy in the Newtonian problem. The qualitative character of the dynamics shows up upon an inspection of diagram A\mathbf{A}. Values of the radius of the universe, aa, greater than a_(max)a_{\max } are not possible. If aa were to become greater than a_(max)a_{\max }, the "potential energy" would exceed the total "energy" and the "kinetic energy" of expansion would have to become negative, which is impossible. Consequently the geometrodynamic system can never be in a state more expanded than a=a_("max ")a=a_{\text {max }}. Starting in a state of small a,(a≪a_(max))a,\left(a \ll a_{\max }\right) and expanding, the universe has for each aa value a perfectly definite da//dtd a / d t value. This velocity of expansion decreases as the expansion proceeds. It falls to zero at the turning point a=a_("max ")a=a_{\text {max }}. Thereafter the system recontracts.
Lack of option is the striking feature of the dynamics. Granted a specific amount of matter [specific MM value in (5)], one has at his disposal no free parameter whatsoever. The value of a_("max ")a_{\text {max }} is uniquely specified by the amount of matter present, and by nothing more. There is no such thing as an "adjustable constant of energy," such as there would have been in a traditional problem of Newtonian dynamics. Where such an adjustable parameter might have appeared in equation (6), there appears instead the fixed number " -1 ." This fixity is the decisive feature of a system bound up into closure. Were one dealing with a collection of rocks out in space, one would have a choice about the amount of dynamite one placed at their center. With a low charge of explosive, one would find the rocks flying out for only a limited distance before gravity halted their flight and brought them to collapse together again. With more propellant, they would fly out with escape velocity and never return. But no such options present themselves here, exactly because Einstein's condition of closure has been imposed; and once closed, always closed. Collapse of the universe is universal. This is simple cosmology in brief.
Einstein's unhappiness at this result was great. At the time he developed general relativity, the permanence of the universe was a fixed item of belief in Western philosophy: "The heavens endure from everlasting to everlasting." Yet the reasoning that led to the fixed equation left open no natural way to change that equation or its fantastic prediction. Therefore Einstein (1917), much against his will, introduced the least unnatural change he could imagine, a so-called cosmological term ($27.11), the whole purpose of which was to avoid the expansion of the universe. A decade later, Hubble (1929) verified the predicted expansion. Thereupon Einstein abandoned the cosmological term, calling it "the biggest blunder of my life" [Einstein (1970)]. Thus ended the first great cycle of apparent contradiction to general relativity, test, and dramatic vindication. Will one ever penetrate the mystery of creation? There is no more inspiring evidence that the answer will someday be "yes" than man's power to predict, and predict correctly, and predict against all expectations, so fantastic a phenomenon as the expansion of the universe.
"Newtonian cosmology" provides an "equation of energy" similar to that of Einstein cosmology, but fails to provide any clean or decisive argument for closure or for the unique constant " -1 ." It considers the mass in any elementary spherical region of space of momentary radius rr, and the gravitational acceleration of a test particle at the boundary of this sphere toward the center of the sphere; thus,
{:(7)(d^(2)r)/(dt^(2))=-((" mass "))/((" distance ")^(2))=-((4pi//3)rhor^(3))/(r^(2))=-(4pi rho)/(3)r.:}\begin{equation*}
\frac{d^{2} r}{d t^{2}}=-\frac{(\text { mass })}{(\text { distance })^{2}}=-\frac{(4 \pi / 3) \rho r^{3}}{r^{2}}=-\frac{4 \pi \rho}{3} r . \tag{7}
\end{equation*}
Consider such imaginary spheres of varied radii drawn in the cosmological medium with the same center. Note that doubling the radius doubles the acceleration. This proportionality between acceleration and distance is compatible with a homogeneous
Box 27.1 (continued)
deceleration of the expansion of the universe. Therefore define an expansion parameter a^(**)a^{*} as the ratio between the radius of any one of these spheres now and the radius of the same sphere at some fiducial instant; thus, a^(**)=r//r_(o)a^{*}=r / r_{o} is to be considered as independent of the particular sphere under consideration. Write rho=rho_(o)r_(o)^(3)//r^(3)\rho=\rho_{o} r_{o}^{3} / r^{3}, where rho_(o)\rho_{o} is the density at the fiducial instant. Insert this expression for rho\rho into the deceleration equation (7), multiply both sides of the equation through by dr//dtd r / d t, integrate, and translate the result from an equation for dr//dtd r / d t to an equation for da^(**)//dtd a^{*} / d t, finding
in agreement with equation (6), except for (1) the trivial differences that arise because a^(**)a^{*} is a dimensionless expansion ratio, whereas aa is an absolute radius with the dimensions of cm , and (2) the all-important difference that here the constant is disposable, whereas in standard Einstein geometrodynamics it has the unique canonical value " -1 ." For more on Newtonian insights into cosmology, see especially Bondi (1961).
Free fall of a particle towards a Newtonian center of attraction according to Newtonian mechanics gives an equation of energy of the same form as (6), except that the "radius of the universe," aa, is replaced by distance, rr, from the center of
attraction. The solution of this problem of free fall is described by a cycloid (diagram B; see also Figure 25.3 and Box 25.4), generated by rolling a circle of diameter a_(max)a_{\max } on a line through an ever increasing angle eta\eta; thus,
Immediately observable today is the present rate of expansion of the universe, with every distance increasing at a rate directly proportional to the magnitude of that distance:*
{:[(((" velocity of recession ")/(" of a galaxy ")))/(" (distance to a galaxy) ")=(" Hubble "constant," "H_(o))∼55km//sec" megaparsec "],[=(1)/(18 xx10^(9)yr)" or "(1)/(1.7 xx10^(28)(cm))],[(10)=(((" rate of increase of the ")/(" radius of the universe itself ")))/(" (radius of the universe) ")=(da//dt)/(a).]:}\begin{align*}
\frac{\binom{\text { velocity of recession }}{\text { of a galaxy }}}{\text { (distance to a galaxy) }} & =\left(\text { Hubble "constant," } H_{o}\right) \sim 55 \mathrm{~km} / \mathrm{sec} \text { megaparsec } \\
& =\frac{1}{18 \times 10^{9} \mathrm{yr}} \text { or } \frac{1}{1.7 \times 10^{28} \mathrm{~cm}} \\
& =\frac{\binom{\text { rate of increase of the }}{\text { radius of the universe itself }}}{\text { (radius of the universe) }}=\frac{d a / d t}{a} . \tag{10}
\end{align*}
The Hubble time, H_(o)^(-1)∼18 xx10^(9)yrH_{o}^{-1} \sim 18 \times 10^{9} \mathrm{yr} (linearly extrapolated back to zero separation on the basis of the expansion rate observed today, as illustrated in the diagram) is predicted to be greater by a factor 1.5 or more (Box 27.3) than the actual time back to the start of the expansion as deduced from the rate of the development of stars ( ∼10 xx10^(9)yr\sim 10 \times 10^{9} \mathrm{yr} ). No such satisfactory concord between prediction and observation on this inequality existed in the 1940's. The scale of distances between galaxy and galaxy in use at that time was short by a factor more than five. The error arose from misidentifications of Cepheid variable stars and of HII regions, which are used as standards of intensity to judge the distance of remote galaxies. The linearly extrapolated time,
back to the start of the expansion was correspondingly short by a factor more than five. The Hubble time came out to be only of the order of 3xx10^(9)yr3 \times 10^{9} \mathrm{yr}. This number obviously violates the inequality
((∼3xx10^(9)yr" Hubble ")/(" time ")) >= 1.5((∼10 xx10^(9)yr;" actual time ")/(" back to start of expansion ")).\binom{\sim 3 \times 10^{9} \mathrm{yr} \text { Hubble }}{\text { time }} \geq 1.5\binom{\sim 10 \times 10^{9} \mathrm{yr} ; \text { actual time }}{\text { back to start of expansion }} .
It implies a curve for dimensions as a function of time not bending down, as in diagram B, but bending up. On some sides the proposal was made to regard the actual curve as rising exponentially. Thus began an era of "theories of continuous creation of matter," all outside the context of Einstein's standard geometrodynamics.
Box 27.1 (continued)
This era ended when, for the first time, the distinction between stellar populations of classes I and II was recognized and as a result Cepheid variables were correctly identified, by Baade (1952,1956)(1952,1956) and when Sandage (1958)(1958) discovered that Hubble had misidentified as bright stars the HII regions in distant galaxies. Then the scale of galactic distances was set straight. Thus ended the second great cycle of an apparent contradiction to general relativity, then test, and then dramatic vindication.
The mystery of the missing matter marks a third cycle of doubt and test with the final decision yet to come. It follows from equation (2) that, if Einstein's closure boundary condition is correct, then the density of mass-energy must exceed a certain lower limit given by the equation
("critical amount of mass-energy required to curve up the geometry of the universe into closure"). A Hubble expansion rate of H_(o)=55km//secH_{o}=55 \mathrm{~km} / \mathrm{sec} Megaparsec implies a lower limit to the density of
as compared to rho∼2xx10^(-31)g//cm^(3)\rho \sim 2 \times 10^{-31} \mathrm{~g} / \mathrm{cm}^{3} of "luminous matter" observed in galaxies ( $29.6\$ 29.6 ) and more being searched for today in the space between the galaxies.
A fuller treatment of cosmology deals with conditions back in the past corresponding to redshifts of 10,000 or more and dimensions 10,000 times less than they are today, when radiation could not be neglected, and even dominated ( $27.10\$ 27.10 ). It also considers even earlier conditions, when anisotropy oscillations of the geometry of the universe as a whole (analogous to the transformation from a cigar to a pancake and back again) may conceivably have dominated (Chapter 30). More broadly, it takes up the evolution of the universe into its present state (Chapter 28) and the present state and future evolution of the universe (Chapter 29). The present chapter examines the basic assumptions that underlie the simple standard cosmology thus traced out, and $27.11\$ 27.11 examines what kinds of qualitative changes would result if one or another of these assumptions were to be relaxed.
§27.2. STRESS-ENERGY CONTENT OF THE UNIVERSETHE FLUID IDEALIZATION
By taking the large-scale viewpoint, one can treat galaxies as "particles" of a "gas" that fills the universe. These particles have internal structure (stars, globular clusters, etc.); but one ignores it. The "particles" cluster on a small scale (clusters of galaxies of size <= 3xx10^(7)\leq 3 \times 10^{7} light years); but one ignores the clustering. To simplify calculations, one even ignores the particulate nature of the "gas" [though one can take it into account, if one wishes, by adopting a kinetic-theory description; see §22.6\S 22.6 for kinetic theory, and Ehlers, Geren, and Sachs (1968) for its application to cosmology]. One removes the particulate structure of the gas from view by treating it in the perfectfluid approximation. Thus, one characterizes the gas by a 4-velocity, u\boldsymbol{u} (the 4 -velocity of an observer who sees the galaxies in his neighborhood to have no mean motion), by a density of mass-energy, rho\rho (the smoothed-out density of mass-energy seen in the frame with 4-velocity u\boldsymbol{u}; this includes the rest mass plus kinetic energy of the galaxies in a unit volume, divided by the volume), and by a pressure pp (the kinetic pressure of the galaxies). The stress-energy tensor for this "fluid of galaxies" is the familiar one
{:(27.1)T=(rho+p)u ox u+gp:}\begin{equation*}
\boldsymbol{T}=(\rho+p) \boldsymbol{u} \otimes \boldsymbol{u}+\boldsymbol{g} p \tag{27.1}
\end{equation*}
where g\boldsymbol{g} is the metric tensor.
Astronomical observations reveal that the rest-mass density of the galaxies is much greater than their density of kinetic energy. The typical ordinary velocities of the galaxies-and of stars in them-relative to each other are
At least, these are the ratios today. Very early in the life of the universe, conditions must have been quite different.
The total density of mass-energy, rho\rho, is thus very nearly the rest-mass density of the galaxies, rho_(rm)\rho_{\mathrm{rm}}. Astronomical observations yield for rho_(rm)\rho_{\mathrm{rm}} today
The rest of this chapter, except for Box 27.4, is Track 2.
No earlier track-2 material is needed as preparation for it, but it is needed as preparation for Chapter 29 (Present state and future evolution of the universe).
Idealization of matter in universe as a perfect fluid ('fluid of galaxies')
Large-scale conditions in universe today:
(1) kinetic energy and pressure of stars and galaxies
(2) density of mass in galaxies
(3) cosmic-ray density
(4) density of intergalactic gas
(5) magnetic fields
(6) radiation density
The cosmic microwave radiation
Not all the matter in the universe is tied up in galaxies; there is also matter in cosmic rays, with an averaged-out density of mass-energy
[Delineating more sharply the value of rho_(ig)\rho_{\mathrm{ig}} is one of the most important goals of current cosmological research. For a review of this question as of 1971, see "The mean mass density of the universe," pp. 56-120 in Peebles (1971).] These sources of mass density, and the associated pressures, one can lump together with the galaxies into the "cosmological fluid," with stress-energy tensor (27.1).
Not all the stress-energy in the universe is in the form of matter. There are also magnetic fields, with mean energy density that almost certainly does not exceed the limit
(corresponding to B_("avg ") <= 10^(-6)GB_{\text {avg }} \leq 10^{-6} G ), and radiation (electromagnetic radiation, neutrino radiation, and perhaps gravitational radiation) totaling, one estimates,
The magnetic fields will be ignored in this chapter; they are unimportant for largescale cosmology, except perhaps very near the "big-bang beginning" of the uni-verse-if they existed then. However, the radiation cannot be ignored, for it plays a crucial role.
Most of the radiation density is in the form of "cosmic microwave radiation," which was discovered by Penzias and Wilson (1965) [see also Dicke, Peebles, Roll, and Wilkinson (1965)], and has been studied extensively since then [for a review, see Partridge (1969)]. The evidence is very strong that this cosmic microwave radiation is a remnant of the big-bang beginning of the universe. This interpretation will be accepted here.
The cosmic microwave radiation has just the form one would expect if the earth were enclosed in a box ("black-body cavity") with temperature 2.7 K . The spectrum is a Planck spectrum with this temperature, and the radiation is isotropic [Boughn, Fram, and Partridge (1971)]. Consequently, its pressure and density of mass-energy are given by the formula,
Thermodynamic considerations ( $27.10\$ 27.10 ) suggest that the universe should also be filled with neutrino radiation and perhaps gravitational radiation that have Planck spectra at approximately the same temperature (∼3K)(\sim 3 \mathrm{~K}). However, they are not detectable with today's technology.
To high accuracy ( <= 300km//sec\leq 300 \mathrm{~km} / \mathrm{sec} ) the mean rest frame of the cosmic microwave radiation near Earth is the same as the mean rest frame of the galaxies in the neighborhood of Earth [Boughn, Fram and Partridge (1971)]. Consequently, the radiation can be included, along with the matter, in the idealized cosmological fluid.
Summary: From the large-scale viewpoint, the stress-energy of the universe can be idealized as a perfect fluid with 4-velocity u\boldsymbol{u}, density of mass-energy rho\rho, pressure pp, and stress-energy tensor
The 4 -velocity u\boldsymbol{u} at a given event P\mathscr{P} in spacetime is the mean 4 -velocity of the galaxies near P\mathscr{P}; it is also the 4 -velocity with which one must move in order to measure an isotropic intensity for the cosmic microwave radiation. The density rho\rho is made up of material density (rest mass plus negligible kinetic energy of galaxies; rest mass plus kinetic energy of cosmic rays; rest mass plus thermal energy of intergalactic gas-all "smeared out" over a unit volume), and also of radiation energy density (electromagnetic radiation, neutrino radiation, gravitational radiation). The pressure pp, like the density rho\rho, is due to both matter and radiation. Today the pressure of the matter is much less than its mass-energy density,
§27.3. GEOMETRIC IMPLICATIONS OF HOMOGENEITY AND ISOTROPY
This chapter will idealize the universe to be completely homogeneous and isotropic. This idealization places tight constraints on the geometry of spacetime and on the motion of the cosmological fluid through it. In order to discover these constraints, one must first give precise mathematical meaning to the concepts of homogeneity and isotropy.
Homogeneity means, roughly speaking, that the universe is the same everywhere at a given moment of time. A given moment of what time? Whose time? This is the crucial question that the investigator asks.
In Newtonian theory there is no ambiguity about the concept "a given moment of time." In special relativity there is some ambiguity because of the nonuniversality of simultaneity, but once an inertial reference frame has been specified, the concept becomes precise. In general relativity there are no global inertial frames (unless spacetime is flat); so the concept of "a given moment of time" is completely ambiguous. However, another, more general concept replaces it: the concept of a threedimensional spacelike hypersurface. This hypersurface may impose itself on one's
Summary of fluid idealization of matter in universe
Spacelike hypersurface as generalization of "moment of time"
attention by reason of natural symmetries in the spacetime. Or it may be selected at the whim or convenience of the investigator. He may find it more convenient to explore spacetime here and there than elsewhere, and to push the hypersurface forward accordingly ("many-fingered time"; the dramatically new conception of time that is part of general relativity). At each event on a spacelike hypersurface, there is a local Lorentz frame whose surface of simultaneity coincides locally with the hypersurface. Of course, this Lorentz frame is the one whose 4-velocity is orthogonal to the hypersurface. These Lorentz frames at various events on the hypersurface do not mesh to form a global inertial frame, but their surfaces of simultaneity do mesh to form the spacelike hypersurface itself.
The intuitive phrase "at a given moment of time" translates, in general relativity, into the precise phrase "on a given spacelike hypersurface." The investigator can go further. He can "slice up" the entire spacetime geometry by means of a "oneparameter family" of such spacelike surfaces. He can give the parameter that distinguishes one such slice from the next the name of "time." Such a one-parameter family of slices through spacetime is not required in the Regge calculus of Chapter 42. However, such a "slicing" is a necessity in most other practical methods for analyzing the dynamics of the geometry of the universe (Chapters 21, 30, and 43). The choice of slicing may dissolve away the difficulties of the dynamic analysis or may merely recognize those difficulties. The successive slices of "moments of time" may shine with simplicity or may only do a tortured legalistic bookkeeping for the dynamics. Which is the case depends on whether the typical spacelike hypersurface is distinguished by natural symmetries or, instead, is drawn arbitrarily.
Homogeneity of the universe means, then, that through each event in the universe there passes a spacelike "hypersurface of homogeneity" (physical conditions identical at every event on this hypersurface). At each event on such a hypersurface the density, rho\rho, and pressure, pp, must be the same; and the curvature of spacetime must be the same.
The concept of isotropy must also be made precise. Clearly, the universe cannot look isotropic to all observers. For example, an observer riding on a 10^(20)eV10^{20} \mathrm{eV} cosmic ray will see the matter of the universe rushing toward him from one direction and receding in the opposite direction. Only an observer who is moving with the cosmological fluid can possibly see things as isotropic. One considers such observers in defining isotropy:
Isotropy of the universe means that, at any event, an observer who is "moving with the cosmological fluid" cannot distinguish one of his space directions from the others by any local physical measurement.
Isotropy of the universe actually implies homogeneity; of this one can convince oneself by elementary reasoning (exercise 27.1).
Isotropy guarantees that the world lines of the cosmological fluid are orthogonal to each hypersurface of homogeneity. This one sees as follows. An observer "moving with the fluid" can discover by physical measurements on which hypersurface through a given event conditions are homogeneous. Moreover, he can measure his own ordinary velocity relative to that hypersurface. If that ordinary velocity is nonzero, it provides the observer with a way to distinguish one space direction in
Isotropy implies fluid world lines orthogonal to homogeneous hypersurfaces
"Homogeneity of universe" defined in terms of spacelike hypersurfaces
his rest frame from all others-in violation of isotropy. Thus in an isotropic universe, where the concept of "observer moving with the fluid" makes sense, each such observer must discover that he is at rest relative to the hypersurface of homogeneity. His world line is orthogonal to that hypersurface.
Exercise 27.1. ISOTROPY IMPLIES HOMOGENEITY
EXERCISE
Use elementary thought experiments to show that isotropy of the universe implies homogeneity.
§27.4. COMOVING, SYNCHRONOUS COORDINATE SYSTEMS FOR THE UNIVERSE
The results of the last section enable one to set up special coordinate systems in the spacetime manifold of an isotropic model universe (Figure 27.1). Choose a hypersurface of homogeneity S_(I)S_{I}. To all the events on it assign coordinate time, t_(I)t_{I}. Lay out, in any manner desired, a grid of space coordinates (x^(1),x^(2),x^(3))\left(x^{1}, x^{2}, x^{3}\right) on S_(I)S_{I}. "Propagate" these coordinates off S_(I)S_{I} and throughout all spacetime by means of the world lines of the cosmological fluid. In particular, assign to every event on a given world line the space coordinates (x^(1),x^(2),x^(3))\left(x^{1}, x^{2}, x^{3}\right) at which that world line intersects S_(I)S_{I}. This assignment has a simple consequence. The fluid is always at rest relative to the space coordinates. In other words, the space coordinates are "comoving"; they are merely labels for the world lines of the fluid. For the time coordinate tt of a given event P\mathscr{P}, use the lapse of proper time, int d tau\int d \tau, of P\mathscr{P} from S_(I)S_{I}, as measured along the fluid world line that passes through P\mathscr{P}, plus t_(I)t_{I} ("standard of time" on the initial hypersurface S_(I)S_{I} ); thus,
{:(27.12)t(P)=t_(I)+(int_(S_(I))^(G)d tau)_("along world line of fuid "):}\begin{equation*}
t(\mathscr{P})=t_{I}+\left(\int_{S_{I}}^{\mathscr{G}} d \tau\right)_{\text {along world line of fuid }} \tag{27.12}
\end{equation*}
The surfaces t=t= constant of such a coordinate system will coincide with the hypersurfaces of homogeneity of the universe. This one sees by focusing attention on observations made by two different observers, AA and BB, who move with the fluid along different world lines. At coordinate time t_(I)t_{I} (on S_(I)S_{I} ) the universe looks the same to BB as to AA. Let AA and BB make observations again after their clocks have ticked away the same time interval Delta tau\Delta \tau. Homogeneity of the initial hypersurface S_(I)S_{I}, plus the deterministic nature of Einstein's field equations, guarantees that AA and BB will again see identical physics. (Identical initial conditions on S_(I)S_{I}, plus identical lapses of proper time during which Einstein's equations govern the evolution of the universe near AA and BB, guarantee identical final conditions.) Therefore, after time lapse Delta tau\Delta \tau, AA and BB are again on the same hypersurface of homogeneity-albeit a different
Construction of a "comoving, synchronous" coordinate system for the universe
Figure 27.1.
Comoving, synchronous coordinate system for the universe, as constructed in $27.4\$ 27.4 of the text. Key features of such a coordinate system are as follows (see $$27.4\$ \$ 27.4 and 27.5). (1) The spatial coordinates move with the fluid, and the time coordinate is proper time along the fluid world lines; i.e., the coordinate description of a particular fluid world line is
{:[(x^(1),x^(2),x^(3))=" constant, "x^(0)-=t=tau+" constant. "],[{: uarr[" proper time measured "],[" along world line "]]]:}\begin{aligned}
\left(x^{1}, x^{2}, x^{3}\right)=\text { constant, } x^{0} \equiv t= & \tau+\text { constant. } \\
& \left.\uparrow \begin{array}{l}
\text { proper time measured } \\
\text { along world line }
\end{array}\right]
\end{aligned}
(2) Any surface of constant coordinate time is a hypersurface of homogeneity of the universe. Every such hypersurface is orthogonal to the world lines of all particles of the fluid. (3) The spatial grid on some initial hypersurface S_(I)S_{I} is completely arbitrary. (4) If gamma_(ij)dx^(i)dx^(j)\gamma_{i j} d x^{i} d x^{j} is the metric on the initial hypersurface in terms of its arbitrary coordinates (with gamma_(ij)\gamma_{i j} a function of x^(1),x^(2),x^(3)x^{1}, x^{2}, x^{3} ), then the metric of spacetime in terms of the comoving, synchronous coordinate system is
ds^(2)=-dt^(2)+a^(2)(t)gamma_(ij)dx^(i)dx^(j)d s^{2}=-d t^{2}+a^{2}(t) \gamma_{i j} d x^{i} d x^{j}
Thus, the entire dynamics of the geometry of the universe is embodied in a single function of time, a(t)=a(t)= "expansion factor"; while the shape (but not size) of the hypersurfaces of homogeneity is embodied in the spatial 3-metric gamma_(ij)dx^(i)dx^(j)\gamma_{i j} d x^{i} d x^{j}.
one from S_(I)\mathcal{S}_{I}, where they began. By virtue of definition (27.12) of coordinate time, the time coordinate at the intersection of BB 's world line with this new hypersurface is t=t_(I)+Delta taut=t_{I}+\Delta \tau; and similarly for AA. Moreover, observers AA and BB were arbitrary. Consequently the new hypersurface of homogeneity, like S_(I)S_{I}, is a hypersurface of constant coordinate time. Q.E.D.
Because the hypersurfaces of homogeneity are given by t=t= constant, the basis vectors del//delx^(i)\partial / \partial x^{i} at any given event P\mathscr{P} are tangent to the hypersurface of homogeneity that goes through that event. On the other hand, the time basis vector, del//del t\partial / \partial t, is tangent to the world line of the fluid through P\mathscr{P}, since that world line has x^(i)=x^{i}= constant along it. Consequently, orthogonality of the world line to the hypersurface guarantees orthogonality of del//del t\partial / \partial t to del//delx^(i)\partial / \partial x^{i} :
{:(27.13a)(del//del t)*(del//delx^(i))=0" for "i=1","2","3.:}\begin{equation*}
(\partial / \partial t) \cdot\left(\partial / \partial x^{i}\right)=0 \text { for } i=1,2,3 . \tag{27.13a}
\end{equation*}
The time coordinate has another special property: it measures lapse of proper time along the world lines of the fluid. Because of this, and because del//del t\partial / \partial t is tangent to the world lines, one can write
{:[(27.13b)del//del t=(d//d tau)_("along fluid's world lines ")],[=u","]:}\begin{align*}
\partial / \partial t & =(d / d \tau)_{\text {along fluid's world lines }} \tag{27.13b}\\
& =\boldsymbol{u},
\end{align*}
where u\boldsymbol{u} is the 4-velocity of the "cosmological fluid." The 4-velocity always has unit length,
Conditions (27.13a,c)(27.13 \mathrm{a}, \mathrm{c}) reveal that, in the comoving coordinate frame [where {:g_(alpha beta)-=(del//delx^(alpha))*(del//delx^(beta))]\left.\mathrm{g}_{\alpha \beta} \equiv\left(\partial / \partial x^{\alpha}\right) \cdot\left(\partial / \partial x^{\beta}\right)\right], the line element for spacetime reads
{:(27.14)ds^(2)=-dt^(2)+g_(ij)dx^(i)dx^(j):}\begin{equation*}
d s^{2}=-d t^{2}+g_{i j} d x^{i} d x^{j} \tag{27.14}
\end{equation*}
Any coordinate system in which the line element has this form is said to be "synchronous" (1) because the coordinate time tt measures proper time along the lines of constant x^(i)x^{i} (i.e., g_(tt)=-1g_{t t}=-1 ), and (2) because the surfaces t=t= constant are (locally) surfaces of simultaneity for the observers who move with x^(i)=x^{i}= constant [i.e., g_(ti)=(del//del t)*(del//delx^(i))=0g_{t i}=(\partial / \partial t) \cdot\left(\partial / \partial x^{i}\right)=0 ]; it is also called a "Gaussian normal coordinate system" (cf. Figure 21.6).
A hypersurface of homogeneity, t=t= constant, has a spatial, three-dimensional geometry described by equation (27.14) with dt=0d t=0 :
{:[(27.15)(ds^(2))_("on hypersurface of homogeneity ")=dsigma^(2)],[=[g_(ij)]_(t=" const ")dx^(i)dx^(j).]:}\begin{align*}
\left(d s^{2}\right)_{\text {on hypersurface of homogeneity }} & =d \sigma^{2} \tag{27.15}\\
& =\left[g_{i j}\right]_{t=\text { const }} d x^{i} d x^{j} .
\end{align*}
To know everything about the 3-geometry on each of these hypersurfaces is to know everything about the geometry of spacetime.
Exercise 27.2. SYNCHRONOUS COORDINATES IN GENERAL
In an arbitrary spacetime manifold (not necessarily homogeneous or isotropic), pick an initial spacelike hypersurface S_(I)S_{I}, place an arbitrary coordinate grid on it, eject geodesic world lines orthogonal to it, and give these world lines the coordinates
where tau\tau is proper time along the world line, beginning with tau=0\tau=0 on S_(I)S_{I}. Show that in this coordinate system the metric takes on the synchronous (Gaussian normal) form (27.14).
EXERCISE
Form of the line element in this coordinate system
Proof that, aside from an over-all "expansion factor," all homogeneous hypersurfaces in the universe have the same 3 -geometry
§27.5. THE EXPANSION FACTOR
To determine the 3-geometry, dsigma^(2)=g_(ij)(t,x^(k))dx^(i)dx^(j)d \sigma^{2}=g_{i j}\left(t, x^{k}\right) d x^{i} d x^{j}, of each of the hypersurfaces of homogeneity, split the problem into two parts: (1) the nature of the 3-geometry on an arbitrary initial hypersurface (dealt with in next section); and (2) the evolution of the 3-geometry as time passes, i.e., as attention moves from the initial hypersurface to a subsequent hypersurface, and another, and another, ... (dealt with in this section).
Assume that one knows the initial 3-geometry-i.e., the coefficients in the space part of the metric,
on the initial hypersurface S_(I)S_{I}-in its arbitrary but explicitly chosen coordinate system. What form will the metric coefficients g_(ik)(t,x^(k))g_{i k}\left(t, x^{k}\right) have on the other hypersurfaces of homogeneity? This question is easily answered by the following argument: Consider two adjacent world lines, aa and B\mathscr{B}, of the cosmological fluid, with coordinates (x^(1),x^(2),x^(3))\left(x^{1}, x^{2}, x^{3}\right) and (x^(1)+Deltax^(1),x^(2)+Deltax^(2),x^(3)+Deltax^(3))\left(x^{1}+\Delta x^{1}, x^{2}+\Delta x^{2}, x^{3}+\Delta x^{3}\right). At time t_(I)t_{I} (on surface {:S_(I))\left.S_{I}\right) they are separated by the proper distance
At some later time tt (on surface SS ), they will be separated by some other proper distance Delta sigma(t)\Delta \sigma(t). Isotropy of spacetime guarantees that the ratio of separations Delta sigma(t)//Delta sigma(t_(I))\Delta \sigma(t) / \Delta \sigma\left(t_{I}\right) will be independent of the direction from aa to B\mathscr{B} (no shearing motion of the fluid). For any given direction, the additivity of small separations guarantees that Delta sigma(t)//Delta sigma(t_(I))\Delta \sigma(t) / \Delta \sigma\left(t_{I}\right) will be independent of Delta sigma(t_(I))\Delta \sigma\left(t_{I}\right). Thus Delta sigma(t)//Delta sigma(t_(I))\Delta \sigma(t) / \Delta \sigma\left(t_{I}\right) must be the same for all pairs of world lines near a given world line. Finally, homogeneity guarantees that this scalar ratio will be independent of position on the initial surface S_(I)S_{I}-i.e., independent of x^(1),x^(2),x^(3)x^{1}, x^{2}, x^{3}. Define a(t)a(t) to be this spatially constant ratio,
Thus, a(t)a(t) is the factor by which the separations of world lines expand between time t_(I)t_{I} and time tt. In other words, the function a(t)a(t) is a universal "expansion factor," or "scale factor."
By combining equations (27.17) and (27.18), one obtains for the separation of adjacent world lines at time tt
This corresponds to the spatial metric at time tt,
{:(27.19)dsigma^(2)=a^(2)(t)gamma_(ij)(x^(k))dx^(i)dx^(j)",":}\begin{equation*}
d \sigma^{2}=a^{2}(t) \gamma_{i j}\left(x^{k}\right) d x^{i} d x^{j}, \tag{27.19}
\end{equation*}
and to the spacetime metric,
{:(27.20)ds^(2)=-dt^(2)+a^(2)(t)gamma_(ij)(x^(k))dx^(i)dx^(j).:}\begin{equation*}
d s^{2}=-d t^{2}+a^{2}(t) \gamma_{i j}\left(x^{k}\right) d x^{i} d x^{j} . \tag{27.20}
\end{equation*}
Figure 27.2.
Inflation of a balloon covered with pennies as a model for the expansion of the universe. Each penny AA may well consider itself to be the center of the expansion because the distance from AA to any neighbor BB or CC increases the more the more remote that neighbor was to begin with ("the Hubble relation"). The pennies themselves do not expand (constancy of sun-Earth distance, no expansion of a meter stick, no increase of atomic dimensions). The spacing today between galaxy and galaxy ( ∼10^(6)lyr\sim 10^{6} \mathrm{lyr} ) is roughly ten times the typical dimension of a galaxy ( ∼10^(5)lyr\sim 10^{5} \mathrm{lyr} ).
Notice that the coefficients gamma_(ij)(x^(k))\gamma_{i j}\left(x^{k}\right) describe the shape not only of the initial hypersurface, but also of all other hypersurfaces of homogeneity. All that changes in the geometry from one hypersurface to the next is the scale of distances. All distances between spatial grid points (fluid world lines) expand by the same factor a(t)a(t), leaving the shape of the hypersurface unchanged. This is a consequence of homogeneity and isotropy; and it is precisely true only if the model universe is precisely homogeneous and isotropic.
Of all the disturbing implications of "the expansion of the universe," none is more upsetting to many a student on first encounter than the nonsense of this idea. The universe expands, the distance between one cluster of galaxies and another cluster expands, the distance between sun and earth expands, the length of a meter stick expands, the atom expands? Then how can it make any sense to speak of any expansion at all? Expansion relative to what? Expansion relative to nonsense! Only later does he realize that the atom does not expand, the meter stick does not expand, the distance between sun and earth does not expand. Only distances between clusters of galaxies and greater distances are subject to the expansion. Only at this gigantic scale of averaging does the notion of homogeneity make sense. Not so at smaller distances. No model more quickly illustrates the actual situation than a rubber balloon with pennies affixed to it, each by a drop of glue. As the balloon is inflated (Figure 27.2) the pennies increase their separation one from another but not a single one of them expands! [For mathematical detail see, e.g., Noerdlinger and Petrosian (1971).]
What expands in the universe, and what does not
EXERCISE
Exercise 27.3. ARBITRARINESS IN THE EXPANSION FACTOR
How much arbitrariness is there in the definition of the expansion factor a(t)a(t) ? Civilization AA started long ago at time t_(A)t_{A}. For it, the expansion factor is
Subsequently men planted civilization BB at time t_(B)t_{B} on a planet in a nearby galaxy. [At this time, the expansion factor a_(A)a_{A} had the value a_(A)(t_(B))a_{A}\left(t_{B}\right). Civilization BB defines the expansion factor relative to the time of its own beginning:
At two subsequent events, CC and DD, of which both civilizations are aware, they assign to the universe in their bookkeeping by no means identical expansion factors,
Show that the relative expansion of the model universe in passing from stage CC to stage DD in its evolution is nevertheless the same in the two systems of bookkeeping:
(a_(A)(t_(D)))/(a_(A)(t_(C)))=((" relative expansion ")/(" from "C" to "D))=(a_(B)(t_(D)))/(a_(B)(t_(C)))\frac{a_{A}\left(t_{D}\right)}{a_{A}\left(t_{C}\right)}=\binom{\text { relative expansion }}{\text { from } C \text { to } D}=\frac{a_{B}\left(t_{D}\right)}{a_{B}\left(t_{C}\right)}
§27.6. POSSIBLE 3-GEOMETRIES FOR A HYPERSURFACE OF HOMOGENEITY
Turn now to the 3-geometry gamma_(ij)dx^(i)dx^(j)\gamma_{i j} d x^{i} d x^{j} for the arbitrary initial hypersurface S_(I)S_{I}. This 3-geometry must be homogeneous and isotropic. A close scrutiny of its three-dimensional Riemann curvature must yield no "handles" to distinguish one point on S_(I)S_{I} from any other, or to distinguish one direction at a given point from any other. "No handles" means that ^((3)){ }^{(3)} Riemann must be constructed algebraically from pure numbers and from the only "handle-free" tensors that exist: the 3-metric gamma_(ij)\gamma_{i j} and
Riemann tensor for homogeneous, isotropic hypersurfaces
the three-dimensional Levi-Civita tensor epsi_(ijk)\varepsilon_{i j k}. (All other tensors pick out preferred directions or locations.) One possible expression for ^((3)){ }^{(3)} Riemann is
Trial and error soon convince one that this is the only expression that both has the correct symmetries for a curvature tensor and can be constructed solely from constants, gamma_(ij)\gamma_{i j}, and epsi_(ijk)\varepsilon_{i j k}. Hence, this must be the 3 -curvature of S_(I)S_{I}. [One says that any manifold with a curvature tensor of this form is a manifold of "constant curvature."]
As one might expect, the metric for S_(I)S_{I} is completely determined, up to coordinate transformations, by the form (27.21) of its curvature tensor. (See exercise 27.4 below). With an appropriate choice of coordinates, the metric reads (see exercise 27.5 below),
{:[dsigma^(2)=gamma_(ij)dx^(i)dx^(j)=K^(-1)[dchi^(2)+sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]" if "K > 0","],[(27.22)dsigma^(2)=gamma_(ij)dx^(i)dx^(j)=dchi^(2)+chi^(2)(dtheta^(2)+sin^(2)theta dphi^(2))" if "K=0","],[dsigma^(2)=gamma_(ij)dx^(i)dx^(j)=(-K)^(-1)[dchi^(2)+sinh^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]" if "K < 0.]:}\begin{align*}
& d \sigma^{2}=\gamma_{i j} d x^{i} d x^{j}=K^{-1}\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \text { if } K>0, \\
& d \sigma^{2}=\gamma_{i j} d x^{i} d x^{j}=d \chi^{2}+\chi^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) \text { if } K=0, \tag{27.22}\\
& d \sigma^{2}=\gamma_{i j} d x^{i} d x^{j}=(-K)^{-1}\left[d \chi^{2}+\sinh ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \text { if } K<0 .
\end{align*}
Absorb the factor K^(-1//2)K^{-1 / 2} or (-K)^(-1//2)(-K)^{-1 / 2} into the expansion factor a(t)a(t) [see exercise 27.3], and define the function
{:(27.23){:[Sigma-=sin chi","," if "k-=K//|K|=+1" ("positive spatial curvature"), "],[Sigma-=chi","," if "k-=K=0" ("zero spatial curvature") "],[Sigma-=sinh chi","," if "k-=K//|K|=-1" ("negative spatial curvature"). "]:}:}\begin{array}{ll}
\Sigma \equiv \sin \chi, & \text { if } k \equiv K /|K|=+1 \text { ("positive spatial curvature"), } \\
\Sigma \equiv \chi, & \text { if } k \equiv K=0 \text { ("zero spatial curvature") } \tag{27.23}\\
\Sigma \equiv \sinh \chi, & \text { if } k \equiv K /|K|=-1 \text { ("negative spatial curvature"). }
\end{array}
Thus write the full spacetime geometry in the form
{:[(27.24)ds^(2)=-dt^(2)+a^(2)(t)gamma_(ij)dx^(i)dx^(j)],[gamma_(ij)dx^(i)dx^(j)=dx^(2)+Sigma^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{gather*}
d s^{2}=-d t^{2}+a^{2}(t) \gamma_{i j} d x^{i} d x^{j} \tag{27.24}\\
\gamma_{i j} d x^{i} d x^{j}=d x^{2}+\Sigma^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)
\end{gather*}
and the 3-curvatures of the homogeneous hypersurfaces in the form
Why is the word "renormalization" appropriate? Previously a(t)a(t) was a scale factor describing expansion of linear dimensions relative to the linear dimensions as they stood at some arbitrarily chosen epoch; but the choice of that fiducial epoch was a matter of indifference. Now a(t)a(t) has lost that arbitrariness. It has been normalized so that its value here and now gives the curvature of a spacelike hypersurface of homogeneity here and now. Previously the factor a(t)a(t) was conceived as dimensionless. Now it has the dimensions of a length. This length is called the "radius of the model universe" when the curvature is positive. Even when the curvature is negative one sometimes speaks of a(t)a(t) as a "radius." Only for zero curvature does the normaliza-
Metric for homogeneous, isotropic hypersurfaces: three possibilities-positive, zero, or negative spatial curvature
tion of a(t)a(t) still retain its former arbitrariness. Thus, for zero-curvature, consider two choices for a(t)a(t), one of them a(t)a(t), the other bar(a)(t)=2a(t)\bar{a}(t)=2 a(t). Then with bar(chi)=(1)/(2)chi\bar{\chi}=\frac{1}{2} \chi, one can write proper distances in the three directions of interest with perfect indifference in either of two ways:
{:[([" proper distance "],[" in the direction "],[" of increasing "chi])=a(t)d chi= bar(a)(t)d bar(chi)","],[([" proper distance "],[" in the direction "],[" of increasing "theta])=a(t)chi d theta= bar(a)(t) bar(chi)d theta","],[([" proper distance "],[" in the direction "],[" of increasing "phi])=a(t)chi sin theta d phi= bar(a)(t) bar(chi)sin theta d phi","]:}\begin{aligned}
& \left(\begin{array}{l}
\text { proper distance } \\
\text { in the direction } \\
\text { of increasing } \chi
\end{array}\right)=a(t) d \chi=\bar{a}(t) d \bar{\chi}, \\
& \left(\begin{array}{l}
\text { proper distance } \\
\text { in the direction } \\
\text { of increasing } \theta
\end{array}\right)=a(t) \chi d \theta=\bar{a}(t) \bar{\chi} d \theta, \\
& \left(\begin{array}{l}
\text { proper distance } \\
\text { in the direction } \\
\text { of increasing } \phi
\end{array}\right)=a(t) \chi \sin \theta d \phi=\bar{a}(t) \bar{\chi} \sin \theta d \phi,
\end{aligned}
No such freedom of choice is possible when the model universe is curved, because then the chi\chi 's in the last two lines are replaced by a function, sin chi\sin \chi or sinh chi\sinh \chi, that is not linear in its argument.
Despite the feasibility in principle of determining the absolute value of the "radius" a(t)a(t) of a curved universe, in practice today's accuracy falls short of what is required to do so. Therefore it is appropriate in many contexts to continue to regard a(t)a(t) as a factor of relative expansion, the absolute value of which one tries to keep from entering into any equation exactly because it is difficult to determine. This motivation will account for the way much of the analysis of expansion is carried out in what follows, with calculations arranged to deal with ratios of aa values rather than with absolute aa values.
Box 27.2 explores and elucidates the geometry of a hypersurface of homogeneity.
EXERCISES
Exercise 27.4. UNIQUENESS OF METRIC FOR 3-SURFACE OF CONSTANT CURVATURE
Let gamma_(ij)\gamma_{i j} and gamma_(i^(')j^('))\gamma_{i^{\prime} j^{\prime}} be two sets of metric coefficients, in coordinate systems {x^(i)}\left\{x^{i}\right\} and {x^(i^('))}\left\{x^{i^{\prime}}\right\}, that have Riemann curvature tensors [derived by equations (8.22) and (8.42)] of the constantcurvature type (27.21). Let it be given in addition that the curvature parameters KK and K^(')K^{\prime} are equal. Show that gamma_(ij)\gamma_{i j} and gamma_(i^(')j^('))\gamma_{i^{\prime} j^{\prime}} are related by a coordinate transformation. [For a solution, see §8.10\S 8.10 of Robertson and Noonan (1968), or §§10\S \S 10 and 27 of Eisenhart (1926).]
Exercise 27.5. METRIC FOR 3-SURFACE OF CONSTANT CURVATURE
(a) Show that the following metric has expression (27.21) as its curvature tensor
(b) By transforming to spherical coordinates (R,theta,phi)(R, \theta, \phi) and then changing to a Schwarzschild radial coordinate ( 2pi r=2 \pi r= "proper circumference"), transform this metric into the form
$$
{:(27.27)dsigma^(2)=(dr^(2))/(1-Kr^(2))+r^(2)(dtheta^(2)+sin^(2)theta dphi^(2)):}\begin{equation*}
d \sigma^{2}=\frac{d r^{2}}{1-K r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) \tag{27.27}
\end{equation*}
$$
(c) Find a further change of radial coordinate that brings the metric into the form (27.22).
Exercise 27.6. PROPERTIES OF THE 3-SURFACES
Verify all statements made in Box 27.2.
Exercise 27.7. ISOTROPY IMPLIES HOMOGENEITY
Use the contracted Bianchi identity ^((3))G^(ik)_(∣k)=0{ }^{(3)} G^{i k}{ }_{\mid k}=0 (where the stroke indicates a covariant derivative based on the 3-geometry alone) to show (1) that ^((3))grad K=0{ }^{(3)} \nabla K=0 in equation (27.21), and therefore to show (2) that direction-independence of the curvature [isotropy; curvature of form (27.21)] implies and demands homogeneity ( KK constant in space).
Box 27.2 THE 3-GEOMETRY OF HYPERSURFACES OF HOMOGENEITY
A. Universe with Positive Spatial
Curvature "'spatially Closed Universe")
Metric of each hypersurface is
{:(1)dsigma^(2)=a^(2)[dchi^(2)+sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{equation*}
d \sigma^{2}=a^{2}\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{1}
\end{equation*}
To visualize this 3-geometry, imagine embedding it in a four-dimensional Euclidean space (such embedding possible here; not possible for general three-dimensional manifold; only four freely disposable functions [w,x,y,z][w, x, y, z] of three variables [alpha,beta,gamma][\alpha, \beta, \gamma] are at one's disposal to try to reproduce six prescribed functions [g_(mn)(alpha,beta,gamma)]\left[g_{m n}(\alpha, \beta, \gamma)\right] of those same three variables).
The embedding is achieved by
{:[w=a cos chi","quad z=a sin chi cos theta","],[(2)x=a sin chi sin theta cos phi","],[y=a sin chi sin theta sin phi","]:}\begin{gather*}
w=a \cos \chi, \quad z=a \sin \chi \cos \theta, \\
x=a \sin \chi \sin \theta \cos \phi, \tag{2}\\
y=a \sin \chi \sin \theta \sin \phi,
\end{gather*}
since it follows that
{:[dsigma^(2)-=dw^(2)+dx^(2)+dy^(2)+dz^(2)],[(3)=a^(2)[dchi^(2)+sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]]:}\begin{align*}
d \sigma^{2} & \equiv d w^{2}+d x^{2}+d y^{2}+d z^{2} \\
& =a^{2}\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{3}
\end{align*}
A 3-surface of positive curvature embedded in four-dimensional Euclidean space. One rotational degree of freedom is suppressed by setting phi=0\phi=0 and pi\pi ("slice through pole," 3sphere in 4 -space looks like a 2 -sphere in 3 -space).
i.e., the surface is a 3 -dimensional sphere in 4dimensional Euclidean space.
To verify homogeneity and isotropy, one need only notice that rotations in the four-dimensional embedding space can move any given point [any given ( w,x,y,zw, x, y, z ) on the 3 -sphere] and any given direction at that point into any other point and direction-while leaving unchanged the line element
The above equations and the picture show that
(1) The 2 -surfaces of fixed chi\chi (which look like circles in the picture, because one rotational degree of freedom is suppressed) are actually 2 -spheres of surface area 4pia^(2)sin^(2)chi4 \pi a^{2} \sin ^{2} \chi; and (theta,phi)(\theta, \phi) are standard spherical coordinates on these 2-spheres.
(2) As chi\chi ranges from 0 to pi\pi, one moves outward from the "north pole" of the hypersurface, through successive 2 -spheres ("shells") of area 4pia^(2)sin^(2)chi4 \pi a^{2} \sin ^{2} \chi (2-spheres look like circles in picture). The area of these shells increases rapidly at first and then more slowly as one approaches the "equator" of the hypersurface, chi=pi//2\chi=\pi / 2. Beyond the equator the area decreases slowly at first, and then more rapidly as one approaches the "south pole", (chi=pi(\chi=\pi; area =0=0 ).
(3) The entire hypersurface is swept out by
( phi\phi is cyclic; phi=0\phi=0 is same as phi=2pi\phi=2 \pi );
its 3 -volume is
{:[V=int(ad chi)(a sin chi d theta)(a sin chi sin theta d phi)],[(5)=int_(0)^(pi)4pia^(2)sin^(2)chi(ad chi)=2pi^(2)a^(3)]:}\begin{align*}
\mathscr{V} & =\int(a d \chi)(a \sin \chi d \theta)(a \sin \chi \sin \theta d \phi) \\
& =\int_{0}^{\pi} 4 \pi a^{2} \sin ^{2} \chi(a d \chi)=2 \pi^{2} a^{3} \tag{5}
\end{align*}
B. Universe with Zero Spatial Curvature ('Spatially Flat Universe'")
Metric of each hypersurface is
{:(6)dsigma^(2)=a^(2)[dchi^(2)+chi^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{equation*}
d \sigma^{2}=a^{2}\left[d \chi^{2}+\chi^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{6}
\end{equation*}
This is a perfectly flat, three-dimensional, Euclidean space described in spherical coordinates. In Cartesian coordinates
{:[x=a chi sin theta cos phi","],[(7)y=a_(chi)sin theta sin phi],[z=a_(chi)cos theta","]:}\begin{gather*}
x=a \chi \sin \theta \cos \phi, \\
y=a_{\chi} \sin \theta \sin \phi \tag{7}\\
z=a_{\chi} \cos \theta,
\end{gather*}
the metric is
{:(8)dsigma^(2)=dx^(2)+dy^(2)+dz^(2):}\begin{equation*}
d \sigma^{2}=d x^{2}+d y^{2}+d z^{2} \tag{8}
\end{equation*}
C. Universe with Negative Spatial Curvature (''Spatially open Universe')
Metric of each hypersurface is
{:(10)dsigma^(2)=a^(2)[dchi^(2)+sinh^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{equation*}
d \sigma^{2}=a^{2}\left[d \chi^{2}+\sinh ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{10}
\end{equation*}
This 3-geometry cannot be embedded in a fourdimensional Euclidean space; but it can be embedded in a flat Minkowski space
{:(11)dsigma^(2)=-dw^(2)+dx^(2)+dy^(2)+dz^(2):}\begin{equation*}
d \sigma^{2}=-d w^{2}+d x^{2}+d y^{2}+d z^{2} \tag{11}
\end{equation*}
To achieve the embedding, set
{:[w=a cosh chi","quad z=a sinh chi cos theta],[(12)x=a sinh chi sin theta cos phi],[y=a sinh chi sin theta sin phi]:}\begin{gather*}
w=a \cosh \chi, \quad z=a \sinh \chi \cos \theta \\
x=a \sinh \chi \sin \theta \cos \phi \tag{12}\\
y=a \sinh \chi \sin \theta \sin \phi
\end{gather*}
insert this into equation (11), and thereby obtain (10).
Equations (12) for the embedded surface imply that
i.e., the surface is a three-dimensional hyperboloid in four-dimensional Minkowski space. (It has the
A 3-surface of negative curvature embedded in four-dimensional Minkowski space. One rotational degree of freedom is suppressed by setting phi=0\phi=0 and pi\pi ("slice through pole"; 3hyperboloid in 4 -space looks like 2 -hyperboloid in 3-space).
same form as a mass hyperboloid in momentum space; see Box 22.5.)
To verify homogeneity and isotropy, one need only notice that "Lorentz transformations" in the embedding space can move any given point on the 3-hyperboloid and any direction through that point into any other point and direction-while leaving unchanged the line element
The above equations and the picture show that
(1) The 2-surfaces of fixed chi\chi (which look like circles in the picture because one rotational degree of freedom is suppressed) are actually 2 -spheres of surface area 4pia^(2)sinh^(2)chi4 \pi a^{2} \sinh ^{2} \chi; and theta\theta, phi\phi ) are standard spherical coordinates on these 2 -spheres.
(2) As chi\chi ranges from 0 to oo\infty, one moves outward from the (arbitrarily chosen) "pole" of the hypersurface, through successive 2 -spheres ("shells") of ever increasing area 4pia^(2)sinh^(2)chi4 \pi a^{2} \sinh ^{2} \chi. For large chi\chi, surface area increases far more rapidly than it would if the hypersurface were flat
( phi\phi is cyclic; phi=0\phi=0 is same as phi=2pi\phi=2 \pi ).
The volume of the hypersurface is infinite.
D. Nonuniqueness of Topology
Warning: Although the demand for homogeneity and isotropy determines completely the local geometric properties of a hypersurface of homogeneity up to the single disposable factor KK, it leaves the global topology of the hypersurface undetermined. The above choices of topology are the most straightforward. But other choices are possible.
This arbitrariness shows most simply when the hypersurface is flat (k=0)(k=0). Write the full spacetime metric in Cartesian coordinates as
{:(16)ds^(2)=-dt^(2)+a^(2)(t)[dx^(2)+dy^(2)+dz^(2)].:}\begin{equation*}
d s^{2}=-d t^{2}+a^{2}(t)\left[d x^{2}+d y^{2}+d z^{2}\right] . \tag{16}
\end{equation*}
Then take a cube of coordinate edge LL
0 < x < L,quad0 < y < L,quad0 < z < L,0<x<L, \quad 0<y<L, \quad 0<z<L,
and identify opposite faces (process similar to rolling up a sheet of paper into a tube and gluing its edges together; see last three paragraphs of §11.5 for detailed discussion). The resulting geometry is still described by the line element (16), but now all three spatial coordinates are "cyclic," like the phi\phi coordinate of a spherical coordinate system:
{:[(t","x","y","z)" is the same event as "],[quad(t","x+L","y+L","z+L).]:}\begin{aligned}
& (t, x, y, z) \text { is the same event as } \\
& \quad(t, x+L, y+L, z+L) .
\end{aligned}
The homogeneous hypersurfaces are now " 3 -toruses" of finite volume
V=a^(3)L^(3),V=a^{3} L^{3},
analogous to the 3 -toruses which one meets under the name "periodic boundary conditions" when analyzing electron waves and acoustic waves in solids and electromagnetic waves in space.
Another example: The 3 -sphere described in part A above (case of "positive curvature") has the same geometry, but not the same topology, as the manifold of the rotation group, SO(3)S O(3) [see exercises 9.12, 9.13, 10.16, and 11.12]. For detailed discussion, see for example Weyl (1946), Coxeter (1963), and Auslander and Markus (1959).
§27.7. EQUATIONS OF MOTION FOR THE FLUID
After the above analysis of any one hypersurface of homogeneity, return to the dynamics of the universe. Examine, first, the evolution of the fluid, as governed by the law grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0.
Recall ( $22.3\$ 22.3 and 23.5) that for a perfect fluid the equations of motion split into two parts. The component along the 4 -velocity, u*(grad*T)=0\boldsymbol{u} \cdot(\boldsymbol{\nabla} \cdot \boldsymbol{T})=0, reproduces the first law of thermodynamics
{:(27.28a)(d//d tau)(rho V)=-p(dV//d tau)",":}\begin{equation*}
(d / d \tau)(\rho V)=-p(d V / d \tau), \tag{27.28a}
\end{equation*}
where VV is the volume of any fluid element. The part orthogonal to the 4 -velocity, (g+u ox u)*(grad*T)=0(\boldsymbol{g}+\boldsymbol{u} \otimes \boldsymbol{u}) \cdot(\boldsymbol{\nabla} \cdot \boldsymbol{T})=0, gives the force equation ("Euler equation")
{:(27.28b)(rho+p)xx(4"-acceleration ")=-(" component of "grad p" orthogonal to "u).:}\begin{equation*}
(\rho+p) \times(4 \text {-acceleration })=-(\text { component of } \boldsymbol{\nabla} p \text { orthogonal to } \boldsymbol{u}) . \tag{27.28b}
\end{equation*}
For a static star ( $23.5\$ 23.5 ) the first law of thermodynamics was vacuous, but the force equation was crucial. For a homogeneous universe, the converse is true; the force equation is vacuous (no accelerations), but the first law of thermodynamics is crucial.
To see that the force equation is vacuous, notice that isotropy guarantees the vanishing of both sides of equation ( 27.28 b). If either side were nonzero at any event P\mathscr{P}, it would distinguish a direction in the homogeneous hypersurface at P\mathscr{P}.
In applying the first law of thermodynamics (27.28a) to cosmology, divide the density and pressure into contributions due to matter and contributions due to radiation:
First discuss the density of mass-energy. Today rho_(m)( >= 10^(-31)(g)//cm^(3))\rho_{m}\left(\geq 10^{-31} \mathrm{~g} / \mathrm{cm}^{3}\right) dominates over rho_(r)(∼10^(-33)(g)//cm^(3))\rho_{r}\left(\sim 10^{-33} \mathrm{~g} / \mathrm{cm}^{3}\right). Matter did not always dominate. Therefore, one cannot set rho_(r)=0\rho_{r}=0. Now discuss the pressure. During that epoch of the universe when pressure was significant cosmologically, p_(r)p_{r} dominated over p_(m)p_{m}. Consequently, one can neglect p_(m)p_{m} at all times, and one can use the "equation of state" for radiation, p_(r)=(1)/(3)rho_(r)p_{r}=\frac{1}{3} \rho_{r}, to write
When (27.30) is inserted into the first law of thermodynamics (27.28a), it yields the result
{:(27.31)(d//d tau)(rho_(m)V)+(d//d tau)(rho_(r)V)=-(1)/(3)rho_(r)dV//d tau:}\begin{equation*}
(d / d \tau)\left(\rho_{m} V\right)+(d / d \tau)\left(\rho_{r} V\right)=-\frac{1}{3} \rho_{r} d V / d \tau \tag{27.31}
\end{equation*}
One cannot integrate this equation until one knows how mass-energy is fed back and forth between matter and radiation-i.e., until one knows another relationship between rho_(m)V\rho_{m} V and rho_(r)V\rho_{r} V. All estimates indicate that, except in the first few seconds of the life of the universe, the energy exchanged between radiation and matter was
negligible compared to rho_(m)V\rho_{m} V and rho_(r)V\rho_{r} V individually (see §28.1). Under these conditions, equation (27.31) can be split into two parts:
{:(27.32a)(d//d tau)(rho_(m)V)=0",":}\begin{equation*}
(d / d \tau)\left(\rho_{m} V\right)=0, \tag{27.32a}
\end{equation*}
First law of thermodynamics used to express densities of radiation and matter in terms of expansion factor
and
{:(27.32b)(d//d tau)(rho_(r)V)+(1)/(3)rho_(r)dV//d tau=0:}\begin{equation*}
(d / d \tau)\left(\rho_{r} V\right)+\frac{1}{3} \rho_{r} d V / d \tau=0 \tag{27.32b}
\end{equation*}
The solutions are simple:
{:(27.33a)rho_(m)V=" constant (conservation of matter) ":}\begin{equation*}
\rho_{m} V=\text { constant (conservation of matter) } \tag{27.33a}
\end{equation*}
and
{:[(27.33b)rho_(r)V^(4//3)=" const "=(rho_(r))/(V^(-1//3))V((" constancy of number ")/(" of photons "))],[" energy "hc//lambda" of "],[" one photon, up "],[" to a factor of "],[" proportionality "]:}\begin{gather*}
\rho_{r} V^{4 / 3}=\text { const }=\frac{\rho_{r}}{V^{-1 / 3}} V\binom{\text { constancy of number }}{\text { of photons }} \tag{27.33b}\\
\text { energy } h c / \lambda \text { of } \\
\text { one photon, up } \\
\text { to a factor of } \\
\text { proportionality }
\end{gather*}
Now what is VV ? It is the volume of any fluid element. It has the value
for a fluid element with edges Delta chi,Delta theta,Delta phi\Delta \chi, \Delta \theta, \Delta \phi. Here chi,theta,phi\chi, \theta, \phi are constant along each world line of the fluid (comoving coordinates). Therefore the element of hyperspherical solid angle Sigma^(2)sin theta Delta chi Delta theta Delta phi\Sigma^{2} \sin \theta \Delta \chi \Delta \theta \Delta \phi (or pseudohyperspherical solid angle for the model of an open universe) is constant throughout all time for any fluid element. Therefore the volume of the fluid element grows in direct proportion to the cube of the expansion parameter aa; thus,
V//a^(3)=" constant "V / a^{3}=\text { constant }
Combining this result with the constancy of rho_(m)V\rho_{m} V and rho_(r)V^(4//3)\rho_{r} V^{4 / 3}, one sees that
Let rho_(mo)\rho_{m o} be the density of matter today, rho_(ro)\rho_{r o} be the density of radiation today, and a_(o)a_{o} be the expansion factor for the universe today. Then, at any time in the past,
These results were based on two key claims, which will be justified in detail later (Chapter 28): the claim that in the epoch when pressure was important p_(m)p_{m} was much smaller than p_(r)p_{r}; and the claim that exchange of mass-energy between radiation and matter was always negligible (except in the first few seconds after the "creation").
§27.8. THE EINSTEIN FIELD EQUATION
Once the time evolution of the expansion factor, a(t)a(t), is known, one can read off the time evolution of the density and pressure directly from equations (27.35). The density and pressure, in turn, determine how the expansion proceeds in time, via Einstein's field equations. Thus the field equations "close the logic loop" and give one a closed mathematical system from which to determine all three quantities, a(t)a(t), p(t)p(t) and rho(t)\rho(t).
One can readily calculate the components of the Einstein tensor for the model universe using the orthonormal basis one-forms,
(With foresight, one will notice ahead of time that isotropy guarantees the equality G_( hat(chi) hat(chi))=G_( hat(theta) hat(theta))=G_( hat(phi) hat(phi) hat(hat(')))G_{\hat{\chi} \hat{\chi}}=G_{\hat{\theta} \hat{\theta}}=G_{\hat{\phi} \hat{\phi} \hat{\hat{\prime}}}, and similar equalities for the Riemann tensor; and one will calculate only G_( hat(chi) hat(chi))G_{\hat{\chi} \hat{\chi}}, the component that is most easily calculated.)
The basis one-forms, omega^( hat(t)),omega^( hat(x)),omega^( hat(theta)),omega^( hat(phi))\boldsymbol{\omega}^{\hat{t}}, \boldsymbol{\omega}^{\hat{x}}, \boldsymbol{\omega}^{\hat{\theta}}, \boldsymbol{\omega}^{\hat{\phi}}, are the orthonormal basis carried along by an observer who moves with the "cosmological fluid." Consequently, T_( hat(t)t)T_{\hat{t} t} is the mass-energy density, rho\rho, that he measures; T_( hat(j)j)T_{\hat{j} j} is the pressure, p;T_( hat(t) hat(j))p ; T_{\hat{t} \hat{j}} vanishes, because he sees no energy flux (no momentum density); and T_( hat(i) hat(j))T_{\hat{i} \hat{j}} vanishes for i!=ji \neq j because he sees no shear stresses:
Equate the Einstein ("moment of rotation") tensor of equations (27.37) to the stress-energy tensor of equations (27.38). And if one insists, include the so-called " Lambda\Lambda-term" or "cosmological term" in the field equations [Einstein (1970): "the biggest blunder of my life"). Thus obtain two nonvacuous field equations. The first is an
"initial value equation," which relates a_(,t)a_{, t} to aa and rho\rho at any initial moment of time:
The second is a "dynamic equation," which gives the second time-derivative of the expansion factor, and thereby governs the dynamic evolution away from the initial moment of time,
{:(27.39b)2(a_(,tt))/(a)=-((a_(,t))/(a))^(2)-(k)/(a^(2))+ubrace(Lambdaubrace)_("omit ")-8pi p:}\begin{equation*}
2 \frac{a_{, t t}}{a}=-\left(\frac{a_{, t}}{a}\right)^{2}-\frac{k}{a^{2}}+\underbrace{\Lambda}_{\text {omit }}-8 \pi p \tag{27.39b}
\end{equation*}
If (27.39b) is to be compared with anything in Newtonian mechanics, it is to be compared with an equation for acceleration (equation of motion), and in the same spirit (27.39a) is to be compared with a first integral of the equation of motion; that is, an equation of energy. In accordance with this comparison, note that one only has to differentiate (27.39a) and combine it with the relation satisfied by the pressure,
("law of conservation of energy") to get the acceleration equation (27.39b). Without any loss of information, one can therefore ignore the "acceleration equation" or "dynamic equation" (27.39b) henceforth, and work with the analog of an energy equation or what is more properly known as an "initial-value equation" (details of initial-value problem for Track-2 readers in Chapter 21).
What shows up here in the limited context of Friedmann cosmology is appropriately viewed in the wider context of general geometrodynamics. Conservation of energy plus one field equation have just been seen to reproduce the other field equations. Conversely, by accepting both field equations, one can derive the law of conservation of energy in the form just stated. Thus, the very act of writing the field equation G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T} (or, if one insists upon the "cosmological term,"
was encouraged by and founded on the automatic vanishing of the divergence grad*G\boldsymbol{\nabla} \cdot \boldsymbol{G} (or the vanishing of the divergences of G\boldsymbol{G} and g\boldsymbol{g} ), because one knew to begin with that energy and momentum are conserved, grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0. It is not surprising, then, that there should be a redundancy between the conservation law, grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0, and the field equations. Neither is it surprising in the dynamics of the Friedmann universe that one can use what is here the one and only interesting component of the conservation law, plus the one and only interesting initial value component ( G_( hat(it))G_{\hat{i t}} component) of the field equations, to obtain the one and only interesting dynamic component ( G_( hat(chi) hat(x))G_{\hat{\chi} \hat{x}} component) of the field equations.
(1) initial value equation
(2) dynamic equation
Why the dynamic equation is superfluous
Side remarks about initial value equations, dynamic equations, and Bianchi identities in more general contexts
Differential equation for expansion factor
Three choices of time parameter for universe:
(1) proper time, tt
(2) expansion factor, a
In a similar way, in more general problems that lack symmetry, one can always eliminate some of the dynamic field equations, but when gravitational radiation is present, one cannot eliminate them all. The dynamic field equations that cannot be eliminated, even in principle, govern the propagation of the gravitational waves. No gravitational waves are present in a perfectly homogeneous and isotropic cosmological model; its high degree of symmetry-in particular, its spherical (2-sphere!) symmetry about chi=0\chi=0-is incompatible with gravitational waves.
Now turn back from general dynamics to Friedmann cosmology. To determine the time evolution of the expansion factor, aa, insert into the initial-value equation (27.39a) the expression for the density of mass-energy given in (27.35a), and arrive at an equation ready for integration,
When one has completed the integration of this equation for a=a(t)a=a(t), one turns back to equation (27.35a,b)(27.35 \mathrm{a}, \mathrm{b}) to get rho(t)\rho(t) and p(t)p(t), and to expression (27.24)(27.24) to get the geometry,
{:(27.41)ds^(2)=-dt^(2)+a^(2)(t)[dchi^(2)+Sigma^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]",":}\begin{equation*}
d s^{2}=-d t^{2}+a^{2}(t)\left[d \chi^{2}+\Sigma^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right], \tag{27.41}
\end{equation*}
thus completing the solution of the problem.
§27.9. TIME PARAMETERS AND THE HUBBLE CONSTANT
To the analysis of this dynamic problem, many investigators have contributed over the years, beginning with Friedmann himself in 1922. They discovered, among other results, that there are three natural choices of time variable, the one of greatest utility depending on the application that one has at hand.
First is tt, the original time variable. This quantity gives directly proper time elapsed since the start of the expansion. This is the time available for the formation of galaxies. It is also the time during which radioactive decay and other physical processes have been taking place.
Second is a(t)a(t), the expansion factor, which grows with time, which therefore serves to distinguish one phase of the expansion from another, and which consequently can be regarded as a parametric measure of time in its own right. The ratio of a(t)a(t) at two times gives the ratio of the dimensions of the universe (cube root of volume) at those two times. It also gives the ratio (1+z)(1+z) of wavelengths at those two times (see §29.2\S 29.2 ). A knowledge of the red shift, zz, experienced in time past by radiation received today is equivalent to a knowledge of a(t)//a_(o)a(t) / a_{o}, where a_(o)a_{o} is the expansion factor today. Specifically, radiation coming in with z=999z=999 is radiation coming in from a time in the history of the universe when it had 10^(-3)10^{-3} of its present dimensions and 10^(-9)10^{-9} of its present volume. During the interval of time while the expansion
parameter is increasing from aa to a+daa+d a, the lapse of proper time, according to (27.40), is
Third is eta(t)\eta(t), the "arc-parameter measure of time." During the interval of time dtd t, a photon traveling on a hypersphere of radius a(t)a(t) covers an arc measured in radians equal to
{:(27.44)d eta=(dt)/(a(t)).:}\begin{equation*}
d \eta=\frac{d t}{a(t)} . \tag{27.44}
\end{equation*}
When the model universe is open instead of closed, the same parameter lets itself be defined. Only the words "hypersphere" and "arc" have to be replaced by the corresponding words for a flat hypersurface of homogeneity (k=0)(k=0) or a hyperboloidal hypersurface (k=-1)(k=-1). In all three cases, the "arc parameter" is defined by the integral of this expression from the start of the expansion:
thus small values of the "arc parameter time," eta\eta, mean early times; and larger values mean later times. In terms of this "arc-parameter measure of time," the metric takes the form
{:(27.46)ds^(2)=a^(2)(eta)[-deta^(2)+dchi^(2)+Sigma^(2)(dtheta^(2)+sin^(2)theta dphi^(2))].:}\begin{equation*}
d s^{2}=a^{2}(\eta)\left[-d \eta^{2}+d \chi^{2}+\Sigma^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] . \tag{27.46}
\end{equation*}
Let a photon start at the "North Pole" of the 3-sphere ( chi=0\chi=0; any theta\theta and phi\phi ) at the "arc parameter time" eta=eta_(1)\eta=\eta_{1}. Then, by the "arc parameter time" eta=eta_(2)\eta=\eta_{2}, the photon has traveled to a new point on the hypersphere and encountered a new set of particles of the "cosmological fluid." They lie at the hyperpolar angle
chi=eta_(2)-eta_(1).\chi=\eta_{2}-\eta_{1} .
When one makes a spacetime diagram on a piece of paper to show what is happening when an effect propagates from one point to another in the universe, one finds it most convenient to take (1) the space coordinate to be chi\chi (the life histories of distinct particles of the "cosmological fluid" thus being represented by distinct vertical lines), and (2) the time coordinate to be eta\eta (so that photons are described by lines inclined at +-45^(@)\pm 45^{\circ} ). No time parameter is more natural to use than eta\eta when one is tracing
out the course of null geodesics. For an example, see the treatment of the cosmological redshift in §29.2\S 29.2. It also turns out that it is simpler analytically (when Lambda\Lambda is taken to be zero) to give a=a(eta)a=a(\eta) and t=t(eta)t=t(\eta) than to give aa directly as a function of time. Thus one gets the connection between the dimension aa and the "arc-parameter time" eta\eta from the formula
From a knowledge of the dimension aa as a function of this time parameter, one immediately gets proper time itself in terms of this time parameter, from the formula
{:(27.48)dt=a(eta)d eta:}\begin{equation*}
d t=a(\eta) d \eta \tag{27.48}
\end{equation*}
An equation (27.40) for the expansion factor and a choice of parameters for marking out time have now set the stage for a detailed analysis of idealized cosmology, and some of the relevant questions have even been asked: How does the characteristic dimension, aa, of the geometry (radius of 3 -sphere, in the case of closure) change with time? What is the spacetime geometry? How do geodesics, especially null geodesics, travel in this geometry? However, additional questions are equally important: Is the expansion of the universe decelerating and, if so, how fast? How do density and pressure of matter and radiation vary with time? And finally, for the simplest and most immediate tie between theory and observation, what is the expansion rate?
In speaking of expansion rate, one refers to the "Hubble constant," the fractional rate of increase of distances,
{:(27.49)H-=((a^(˙))(t))/(a(t)):}\begin{equation*}
H \equiv \frac{\dot{a}(t)}{a(t)} \tag{27.49}
\end{equation*}
which is normally evaluated today HH (today) -=H_(0)\equiv H_{0}, but is in principle defined as a function of time for every phase of the history of the universe. The reciprocal of HH is the "Hubble time," H^(-1)H^{-1}. This quantity represents the time it would have taken for the galaxies to attain their present separations, starting from a condition of infinite compaction, if they had maintained for all time their present velocities ("time for expansion with dimensions linearly extrapolated back to the start"). For the conversion from astrophysical to geometric units and to years, take the currently accepted value, H_(o)=55km//secH_{o}=55 \mathrm{~km} / \mathrm{sec} megaparsec (Box 29.4), as an illustration:
{:[H_(o)=(55(km)//sec)/((299,793(km)//sec)(3.0856 xx10^(24)(cm)" or "3.2615 xx10^(6)yr" of time "))],[=0.59 xx10^(-28)" per "cm" of light-travel time "],[(27.50)H_(o)^(-1)=1.7 xx10^(28)cm" of light-travel time or "5.6 xx10^(-11)" fractional expansion per "yr","],[10^(9)yr]:}\begin{align*}
H_{o} & =\frac{55 \mathrm{~km} / \mathrm{sec}}{(299,793 \mathrm{~km} / \mathrm{sec})\left(3.0856 \times 10^{24} \mathrm{~cm} \text { or } 3.2615 \times 10^{6} \mathrm{yr} \text { of time }\right)} \\
& =0.59 \times 10^{-28} \text { per } \mathrm{cm} \text { of light-travel time } \\
H_{o}^{-1} & =1.7 \times 10^{28} \mathrm{~cm} \text { of light-travel time or } 5.6 \times 10^{-11} \text { fractional expansion per } \mathrm{yr}, \tag{27.50}\\
& 10^{9} \mathrm{yr}
\end{align*}
§27.10. THE ELEMENTARY FRIEDMANN COSMOLOGY OF A CLOSED UNIVERSE
Take the simplest cosmological model, an isotropic homogeneous closed universe with Lambda=0\Lambda=0, and trace out its features in all detail in the two limiting cases where matter dominates and where radiation dominates. The term "Friedmann universe" is used here for both cases, although the matter-dominated model is sometimes referred to as the Friedmann universe and the radiation-dominated one as the Tolman universe. In this analysis, it will be appropriate to let the variable a(t)a(t) represent the radius of the universe, as measured in cm , because only by reference to this radius does one have the tool in hand to discuss all the interesting geometric effects that in principle lend themselves to observation. After this discussion, it will be enough, in dealing with other models, to summarize their principal parts and comment on their differences from this simple model, without repeating the full investigation. Any reference to an open universe or any so-called "cosmological constant" or its effects will therefore be deferred to a brief final section, §27.11\S 27.11. There the variable a(t)a(t) will sometimes be taken to represent only a parameter of relative expansion, as is appropriate for discussions reaching out only to, say, z=0.1z=0.1, where global geometric issues are not taken up.
Rewrite the controlling component (27.40) of Einstein's field equation in the form
In both cases, the problem lends itself to comparison to the problem of particle motion in Newtonian mechanics with "total energy" -1 and with an "effective potential energy" of the qualitative form shown in diagram A of Box 27.1-apart from minor differences in shape according as the potential goes as -1//a-1 / a or as -1//a^(2)-1 / a^{2}. The principal features of the solution are collected in Box 27.3.
It is a striking feature of the radiation-dominated era of the early Friedmann universe that the density of the radiation depends on time according to a simple universal law,
(final line and final column of Box 27.3). This circumstance may someday provide
Features of a closed Friedmann universe with Lambda=0\Lambda=0 :
(1) radius as function of time 都
Box 27.3 SOLUTIONS FOR THE ELEMENTARY FRIEDMANN COSMOLOGY OF A CLOSED UNIVERSE IN THE TWO LIMITING CASES IN WHICH (1) MATTER DOMINATES AND RADIATION IS NEGLIGIBLE, AND (2) RADIATION DOMINATES AND MATTER IS NEGLIGIBLE
Idealization for
dynamics of 3-sphere
Idealization for
dynamics of 3-sphere| Idealization for |
| :---: |
| dynamics of 3-sphere |
Matter dominated
Radiation dominated
"Idealization for
dynamics of 3-sphere" Matter dominated Radiation dominated| Idealization for <br> dynamics of 3-sphere | Matter dominated | Radiation dominated |
| :---: | :---: | :---: |
Value of constant in this "potential" in terms of conditions at some standard epoch
Solution of dynamic equation expressed parametrically in terms of "arc parameter" eta\eta (radians of arc distance on 3 -sphere covered by a photon travelling ever since start of expansion)
Range of eta\eta from start of expansion to end of recontraction
Nature of curve relating radius aa to
time tt
Hubble time
H^(-1)=(a)/((da//dt))=(a^(2))/((da//d eta))H^{-1}=\frac{a}{(d a / d t)}=\frac{a^{2}}{(d a / d \eta)}
Effective "potential" in
back into past to redshift z∼10,000z \sim 10,000; through today and through phase of maximum expansion, and recontraction down to dimensions ∼10,000\sim 10,000-fold smaller than today V(a)=-(a_(max))/(a)V(a)=-\frac{a_{\max }}{a} V(a)=-(a^(**2))/(a^(2))V(a)=-\frac{a^{* 2}}{a^{2}} a_(max)=(8pi)/(3)a_(o)^(3)rho_("mo ")a_{\max }=\frac{8 \pi}{3} a_{o}^{3} \rho_{\text {mo }} a^(**2)=(8pi)/(3)a_(o)^(4)rho_(ro)a^{* 2}=\frac{8 \pi}{3} a_{o}^{4} \rho_{r o} a=(a_("max "))/(2)(1-cos eta)a=\frac{a_{\text {max }}}{2}(1-\cos \eta) a=a^(**)sin etaa=a^{*} \sin \eta t=(a_(max))/(2)(eta-sin eta)t=\frac{a_{\max }}{2}(\eta-\sin \eta) 2pi2 \pi (one trip around the universe)
cycloid
very early phase of expansion, for redshifts z∼1,000z \sim 1,000 and greater; and corresponding phase in late stages of recontraction; not directly relevant today.
Matter dominated H^(-1) >= 1.5 tH^{-1} \geq 1.5 t H^(-1) >= 2tH^{-1} \geq 2 t
trapolated" time and actual time
back to start of expansion
Density of mass-energy
This density expressed in terms of Hubble expansion rate
Inequality satisfied by density
Analysis of magnification of distant galaxy by curvature of intervening space
a tool to tell how many kinds of radiation contributed to rho_(r)\rho_{r} in the early universe; or, in other words, to learn about field physics from observational cosmology. Express the density of radiation in the form
It would be surprising if electromagnetism made the sole contribution to the radiation density, since the following additional mechanisms are available to sop up thermal energy from a violently radiating source:
{:[" electromagnetic radiation (already considered), ",f_(em)=8;],[" gravitational black body radiation, ",f_(g)=8;]:}\begin{array}{lr}
\text { electromagnetic radiation (already considered), } & f_{e m}=8 ; \\
\text { gravitational black body radiation, } & f_{g}=8 ;
\end{array}
neutrino plus antineutrino radiation of the electron-
neutrino type [its contribution depends on the chemical
potential of the neutrinos, on which see Brill and
Wheeler (1957); a zero value is assumed here for that potential],
f_(ev)=7;f_{e v}=7 ;
neutrino plus antineutrino radiation of the muon-
neutrino type [with the same assumptions as for nu_(e)\nu_{e} 's], quadf_(mu nu)=7\quad f_{\mu \nu}=7;
pairs of positive and negative electrons produced out
of the vacuum when temperatures are of the order of T=mc^(2)//k=0.59 xx10^(10)KT=m c^{2} / k=0.59 \times 10^{10} \mathrm{~K} and higher, evaluated in
the approximation in which these particles are treated as overwhelmingly more numerous than the unpaired electrons that one sees today,
f_(e^(+)e^(-))=14;f_{e^{+} e^{-}}=14 ;
other particles such as mesons created out of the vac-
uum when temperatures are two orders of magnitude
higher ( ∼10^(12)K\sim 10^{12} \mathrm{~K} ), and baryon-antibaryon pairs created
out of the vacuum when temperatures are of the order of ∼10^(13)K\sim 10^{13} \mathrm{~K} and higher,
sum of these ff-values, (f_(mu^(+)mu^(-)),f_(pi),dots;)/()\frac{f_{\mu^{+} \mu^{-}}, f_{\pi}, \ldots ;}{}. (27.55)
As the expansion proceeds and temperatures drop below 10^(13)K10^{13} \mathrm{~K}, then 10^(12)10^{12}, then 10^(10)10^{10}, the various particle pairs presumably annihilate and disappear [see, however, Alfvén and Klein (1962), Alfvén (1971), Klein (1971), and Omnes (1969)]. One is left with the radiations of zero rest mass, and only these radiations, contributing to the specific heat of the vacuum. At the phases of baryon-antibaryon and electronpositron annihilation, the thermal gravitational radiation present has already effectively decoupled itself from the matter, according to all current estimates. Therefore the energy set free by annihilation of matter and antimatter is expected to pour at first into the other two carriers of energy: neutrinos and electromagnetic radiation. However, the neutrinos also decouple early (after baryon-antibaryon annihilation; before full electron-positron annihilation), because the mean free path for neutrinos
rises rapidly with expansion. The energy of the subsequent annihilations goes almost exclusively into electromagnetic radiation. Thus the temperatures of the three radiations at the present time are expected to stand in the order
T_(em)T_{\mathrm{em}} has been measured to be 2.7K;T_(nu)2.7 \mathrm{~K} ; T_{\nu} is calculated to be (4//11)^(1//3)T_(em)=1.9K(4 / 11)^{1 / 3} T_{\mathrm{em}}=1.9 \mathrm{~K}, and T_(g)T_{g} has been calculated to be 1.5 K [Matzner (1968)] in a model where gravitons decouple during an early, quark-dominated era.
Decoupled radiation, once in a Planck spectrum, remains in a Planck spectrum (see Box 29.2). Expansion leaves constant the product rho_(r," decoupled ")a^(4)\rho_{r, \text { decoupled }} a^{4} or the product T_(r," decoupled ")^(4)a^(4)T_{r, \text { decoupled }}^{4} a^{4}. Compare the temperature of this particular radiation now to the temperature of the same radiation at any chosen fiducial time t_("fid ")t_{\text {fid }} after its era of decoupling. Find
{:(27.57)T_(r," fid ")=(a_("now "))/(a_("fid "))T_(r," now ")=(1+z)T_(r," now "):}\begin{equation*}
T_{r, \text { fid }}=\frac{a_{\text {now }}}{a_{\text {fid }}} T_{r, \text { now }}=(1+z) T_{r, \text { now }} \tag{27.57}
\end{equation*}
Here zz represents the red shift of any "tracer" spectral line, given off at the fiducial time, and observed today, relative to the standard wavelength of the same transition as observed in the laboratory.
If the three radiations could be catalyzed into thermodynamic equilibrium, then all radiations could be treated on the same footing during the radiation-dominated era of cosmology. Their individual ff values could be added directly to give f=8+8+7+7=30f=8+8+7+7=30. Temperature and time would then be connected by the formula
This formula together with (27.57) implies the relation
{:(27.58b)[((T_(r" now "))/(10^(10)(K)))(1+z)]^(2)((t_(fid))/(1sec))=1.19.:}\begin{equation*}
\left[\left(\frac{T_{r \text { now }}}{10^{10} \mathrm{~K}}\right)(1+z)\right]^{2}\left(\frac{t_{\mathrm{fid}}}{1 \mathrm{sec}}\right)=1.19 . \tag{27.58b}
\end{equation*}
This relation concerns two radiations: (1) the actual electromagnetic radiation with Planck spectrum (a continuum); and (2) the redshift and time of emission of a "tracer radiation" (a line spectrum). A measured departure from this relation could serve as one potential (indirect) indication that, in accordance with standard theory, neutrinos and gravitational radiation today are cooler than electromagnetic radiation.
Turn now from the radiation-dominated era of cosmology to the matter-dominated era. Numbers sometimes elicit more response from the imagination than formulas. Therefore idealize to a matter-dominated cosmology, and for the moment arbitrarily adopt 20 xx10^(9)yr20 \times 10^{9} \mathrm{yr} and 10 xx10^(9)yr10 \times 10^{9} \mathrm{yr} as Hubble time and actual time, respectively, back to the start of the expansion. It is certain that future work will show both numbers to require revision, but probably not by more than a factor 2 , in the opinion of observational cosmologists. Since any judgment on the best numbers is subject
(3) later era, when matter dominates
to uncertainty, one can pick the numbers to be simple as well as reasonable. From Box 27.3, one then deduces the present value of the arc parameter time eta\eta,
{:(27.60)eta=1.975(" or "113.2^(@)):}\begin{equation*}
\eta=1.975\left(\text { or } 113.2^{\circ}\right) \tag{27.60}
\end{equation*}
(arc traveled by a photon on the 3 -sphere from the start of the expansion to today.) This fixed, all other numbers emerge as shown in Box 27.4.
Box 27.4 A TYPICAL COSMOLOGICAL MODEL COMPATIBLE WITH ASTRONOMICAL OBSERVATIONS AND WITH EINSTEIN'S CONCEPTION OF COSMOLOGY ( Lambda=0\Lambda=0; Universe Closed)
Radius at phase of maximum expansion,
Time from start to maximum,
Time from start to final recontraction,
Time from start to today (adopted value),
Radius today,
Hubble time today (adopted value),
Hubble expansion rate today,
Deceleration parameter today, q_(o)q_{o} [equation (29.1b)]
Density today (3//8pia_(o)^(2))+(3H_(o)^(2)//8pi)\left(3 / 8 \pi a_{o}{ }^{2}\right)+\left(3 H_{o}^{2} / 8 \pi\right),
Volume today, 2pi^(2)a_(o)^(3)2 \pi^{2} a_{o}{ }^{3},
Density at maximum (3//8pia^(2))+(3H^(2)//8pi)\left(3 / 8 \pi a^{2}\right)+\left(3 H^{2} / 8 \pi\right),
It must be emphasized that these numbers do not deserve the title of "canonical," however convenient that adjective may be for describing them; they can at most be called illustrative.
Figure 27.3.
Many Schwarzschild zones are fitted together to make a closed universe. This universe is dynamic because a test particle at the interface between two zones rises up against the gravitational attraction of each and falls back under the gravitational attraction of each. Therefore the two centers themselves have to move apart and move back together again. The same being true for all other pairs of centers, it follows that the lattice universe itself expands and recontracts, even though each Schwarzschild geometry individually is viewed as static. This diagram is taken from Lindquist and Wheeler (1957).
If every five seconds a volume of space is added to the universe, a volume equivalent to a cube 10^(5)lyr(=0.95 xx10^(23)(cm))10^{5} \mathrm{lyr}\left(=0.95 \times 10^{23} \mathrm{~cm}\right) on an edge, about equal to the volume occupied by the Milky Way, where does that volume make its entry? Rather than look for an answer, one had better reexamine the question. Space is not like water. The outpouring of fresh water beneath the ocean at the Jesuit Spring off Mount Desert Island can be detected and measured by surrounding the site with flowmeters. There is no such thing as a flowmeter to tell "how fast space is streaming past." The very idea that "space flows" is mistaken. There is no way to define a flow of space, not least because there is no way to measure a flow of space. Water, yes; space, no. Life is very different for the flowmeter, according as it is stationary or moving with respect to the water. For a particle in empty space, however, physics is indistinguishable regardless of whether the particle is at rest or moves at high velocity relative to some chosen inertial frame. To try to pinpoint where those cubic kilometers of space get born is a mistaken idea, because it is a meaningless idea.
One can get a fresh perspective on what is going on in expansion and recontraction by turning from a homogeneous isotropic closed universe to a Schwarzschild lattice closed universe. [Lindquist and Wheeler (1957)]. In the former case, the mass is idealized as distributed uniformly. In the latter, the mass is concentrated into 120 identical Schwarzschild black holes. Each is located at the center of its own cell, of dodecahedral shape, bounded by 12 faces, each approximately a pentagon; and space is empty. The dynamics is easy to analyze in the approximation in which each lattice cell is idealized as spherical, a type of treatment long familiar in solid-state physics as the "Wigner-Seitz approximation" (references in Lindquist and Wheeler). In this approximation, the geometry inside each lattice cell is treated as having exactly the Schwarzschild character (Figure 27.3); a test particle placed midway between black hole A and black hole B rises against the attraction of each, and ultimately falls back toward each, according to the law developed in Chapter 25 [equation (25.28) with a shift of pi\pi in the starting point for defining eta\eta ],
Accordingly, the two masses in question must fall toward each other; and so it is with all the masses. One comes out in this way with the conclusion that the lattice
(4) "Where is the new space created during expansion?'"-a meaningless question
universe follows the same law of expansion and recontraction as the Friedmann universe to an accuracy of better than 4 per cent [Lindquist and Wheeler; Wheeler (1964a), pp. 370-381]. Now ask again the same meaningless question about where the cubic kilometers of space pour into the universe while it is expanding, and where they pour out while it is recontracting. Receive a fuller picture why the question is meaningless. Surrounding each center of mass, the geometry is and remains the Schwarzschild geometry (until eventually the black holes come so close together that they coalesce). The situation inside each cell is therefore static. Moreover, the interface between cell and cell is defined in imagination by a sprinkling of test particles so light that they have no influence on the geometry or its dynamics. The matchup between the geometry in one cell and the next is smooth ("tangency between the two geometries"). There is nothing abnormal whatsoever in the spacetime on and near the interface. One has as little right to say those cubic kilometers are "created", here as anywhere else. To speak of the "creation" of space is a bad way of speaking, and the original question is a bad question. The right way of speaking is to speak of a dynamic geometry. So much for one question!
In charting the dynamics of the geometry of a Friedmann universe, one often finds that it simplifies things to take as space coordinate the hyperpolar angle chi\chi, measured from some chosen world line (moving with the "cosmological fluid") as standard of reference; and to take as time coordinate the arc-parameter measure of time, eta\eta, as illustrated in Figure 27.4.
Inspection of the (chi,eta)(\chi, \eta)-diagram makes it clear that photons emitted from matter at one point cannot reach, in a limited time, any matter except that which is located in a limited fraction of the 3 -sphere. In a short time tt, according to Box 27.3, a photon can cover an arc distance on the 3 -sphere equal only to eta=(2t//a^(**))^(1//2)\eta=\left(2 t / a^{*}\right)^{1 / 2}. Moreover, what is true of photons is true of other fields, forces, pressures, energies and influences: they cannot reach beyond this limit. Evidently the 3 -sphere at time tt is divided into a number of "zones," N=((" number of ")/(" "zones" "))=(((" hyperspherical solid ")/(" angle of entire 3-sphere ")))/(((" hyperspherical solid ")/(" angle of one zone ")))=(2pi^(2))/(4pichi^(3//3))=(3pi)/(2^(5//2))((a^(**))/(t))^(3//2)N=\binom{\text { number of }}{\text { "zones" }}=\frac{\binom{\text { hyperspherical solid }}{\text { angle of entire 3-sphere }}}{\binom{\text { hyperspherical solid }}{\text { angle of one zone }}}=\frac{2 \pi^{2}}{4 \pi \chi^{3 / 3}}=\frac{3 \pi}{2^{5 / 2}}\left(\frac{a^{*}}{t}\right)^{3 / 2},
effectively decoupled one from the other. As time goes on, there are fewer separate zones, and ultimately every particle has been subjected to influences from every other particle in the model universe.
EXERCISES
Exercise 27.8. MATTER-DOMINATED AND RADIATION-DOMINATED REGIMES OF FRIEDMANN COSMOLOGY
Derive the results listed in the last two columns of Box 27.3, except for the focusing properties of the curved space.
Figure 27.4.
Use of "are parameter" eta\eta as a time coordinate and hyperpolar angle chi\chi as a space coordinate to describe travel of a photon ( +-45^(@)\pm 45^{\circ} line) in a Friedmann universe that is matter-dominated (center) or radiationdominated (right). The burst of photons is emitted from the " NN-pole" of the 3 -sphere at a time very little after the big bang, and the locus of the cloud of photons at subsequent stages of the expansion and recontraction is indicated by sections of the 3 -sphere in the diagrams at the left. The matter-dominated Friedmann universe appears to be a reasonable model for the physical universe, except when its dimensions have fallen to the order of one ten-thousandth of those at maximum expansion or less ("radiation regime").
Exercise 27.9. TRANSITION FROM RADIATION-DOMINATED REGIME TO MATTER-DOMINATED REGIME
Including both the radiation and the matter terms in equation (27.51), restate the equation in terms of the arc parameter eta\eta (with d eta=dt//ad \eta=d t / a ) as independent variable, and integrate to find
(a) Verify that under appropriate conditions these expressions reduce at early times to a "circle" relation between radius and time and to a "cycloid" relation later.
(b) Assign to a^(**2)a^{* 2} the value a_(o)a_(max)//10,000a_{o} a_{\max } / 10,000 (why?) and construct curves for the dimensionless measures of density,
What conclusions emerge from inspecting the logarithmic slope of these curves?
Exercise 27.10. THE EXPANDING AND RECONTRACTING SPHERICAL WAVE FRONT
An explosion takes place at the " NN-pole" of the matter-dominated Friedmann model universe at the value of the "arc parameter time" eta=pi//3\eta=\pi / 3, when the radius of the universe has reached half its peak value. The photons from the explosion race out on a spherical wave front. Through what fraction of the "cosmological fluid" has this wave front penetrated at that instant when the wave front has its largest proper surface area?
§27.11. HOMOGENEOUS ISOTROPIC MODEL UNIVERSES THAT VIOLATE EINSTEIN'S CONCEPTION OF COSMOLOGY
Open Friedmann universe with Lambda=0\Lambda=0 :
(1) expansion factor as function of time
(2) early stage-same as for closed universe
It violates Einstein's conception of cosmology (Box 27.1)-though not the equations of his theory-to replace the closed 3-sphere of radius aa by the open hyperboloidal geometry of equation (27.22) with the same scale length aa. Even so, the results of Box 27.3 continue to apply in the two limiting regimes of matter-dominated and radiation-dominated dynamics when the following changes are made. (1) Change the constant -1 on the righthand side of the analog of a "Newtonian energy equation" to +1 , thus going over from a bound system (maximum expansion) to an open system (forever expanding). (2) Replace (1-cos eta(1-\cos \eta ) by (cosh eta-1)(\cosh \eta-1), sin eta\sin \eta by sinh eta,cos eta\sinh \eta, \cos \eta by cosh eta\cosh \eta, and (eta-sin eta)(\eta-\sin \eta) by (sinh eta-eta)(\sinh \eta-\eta). (3) The range of the "arc parameter" eta\eta now extends from 0 to oo\infty, and the curve relating "radius" aa to time tt changes from cycloid or circle to an ever-rising curve. (4) The listed inequalities on the Hubble time (as related to the actual time of expansion) and on the density (as related to 3H_(o)^(2)//8pi3 H_{o}^{2} / 8 \pi ) no longer hold. (5) The formulas given in Box 27.3 for conditions at early times continue to hold, for a simple reason: at early times the curvature of spacetime "in the direction of increasing time" [the extrinsic curvature (6//a^(2))(da//dt)^(2)\left(6 / a^{2}\right)(d a / d t)^{2} as it appears in Box 27.1, equation (2)] is overwhelmingly more important than the curvature within any hypersurface of homogeneity, +-6//a^(2)\pm 6 / a^{2} (the intrinsic curvature); therefore it makes no detectable difference at early times whether the sign is plus or minus, whether the space is closed or open, or whether the geometry of space is spherical or hyperboloidal.
Why doesn't it make a difference? Not why mathematically, but why physically, doesn't it make a difference in early days whether the space is open or closed?
Because photons, signals, pressures, forces, and energies cannot get far enough to "smell out" the difference between closure and openness. The "zones of influence" of (27.62) are too small for any one by itself to sense or to respond significantly to any difference between a negative space curvature -6//a^(2)-6 / a^{2} and a positive space curvature +6//a^(2)+6 / a^{2}. Therefore the simple power-law time-dependence of the density of the mass-energy of radiation given in Box 27.3 for a closed universe holds equally well in the earliest days of a radiation-dominated, open, isotropic model universe; thus,
Only at a later stage of the expansion, when the "extrinsic curvature" term [equation (2), Box 27.1], (6/a {:a^(2))(da//dt)^(2)\left.a^{2}\right)(d a / d t)^{2} (initially varying as 1.5t^(-2)1.5 t^{-2}, according to Box 27.3) has fallen to a value of the same order of magnitude as the "intrinsic curvature" term +-6//a^(2)\pm 6 / a^{2} (initially varying as +-3a^(**-1)t^(-1)\pm 3 a^{*-1} t^{-1} ), does the sign of the intrinsic curvature begin to matter. Only then do the differences in rate of expansion begin to show up that distinguish the open model universe from the closed one.
The open model goes on expanding forever. Therefore the density of mass-energy, whether matter-dominated and proportional to a_(max)//a^(3)a_{\max } / a^{3}, or radiation-dominated and proportional to a^(**2)//a^(4)a^{* 2} / a^{4}, or some combination of the two, (1) ultimately falls to a level that is negligible in comparison with the intrinsic curvature, -6//a^(2)-6 / a^{2}, and (2) thereafter can be neglected. Under these circumstances, the only term left to balance the intrinsic curvature is the extrinsic curvature. The important component of the field equation (after removal from all terms of a common factor 3) now reads
For a closed universe, the two terms (one sixth the extrinsic curvature and one sixth the intrinsic curvature) have the same sign, and any equation like (27.67) leads to an impossibility. Here, however, rather than impossibility, one has the remarkably simple solution
and find that (27.69), solution as it is of Einstein's empty-space field equation, is identical with the Lorentz-Minkowski metric of flat spacetime,
{:(27.71)ds^(2)=-dt_("new ")^(2)+dr^(2)+r^(2)(dtheta^(2)+sin^(2)theta dphi^(2)):}\begin{equation*}
d s^{2}=-d t_{\text {new }}^{2}+d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) \tag{27.71}
\end{equation*}
(see Box 27.2C). This geometry had acquired the flavor of an expanding universe
(3) late stage-expansion forever
Homogeneous cosmologies with Lambda!=0\Lambda \neq 0 :
(1) equation for evolution of expansion factor
(2) qualitative features of evolution
because the cosmological fluid, too thinly spread to influence the dynamics of the geometry, and serving only to provide marker points, was flying out in all directions [for a fuller discussion of this "expanding Minkowski universe," see, for example, Chapter 16 of Robertson and Noonan (1968)]. The typical spacelike hypersurface of homogeneity looks to have a curved 3-space geometry, and does have a curved geometry (instrinsic curvature), because the slice (27.70) through flat spacetime is itself curved (extrinsic curvature).
Turn now to a second violation of Einstein's conception of cosmology: a cosmological term in the field equation (27.39),
In analyzing the implications of this broadened equation, turn attention from the "radius" a(t)a(t) itself, which was the focus of interest in the previous section, §27.10\S 27.10, on Friedmann cosmology. Recognize that present measurements have not yet provided a good, direct handle on the absolute dimension a(t)a(t) of the universe. However, they do give good figures for the redshift zz and therefore for the ratio between aa at the time of emission and a=a_(o)a=a_{o} now,
For any comparison with observations designed to fix limits on kk (Einstein value: k=+1k=+1 ) and on Lambda\Lambda (expected to be zero), it is therefore appropriate to rewrite the foregoing equations so that they refer as much as possible only to ratios. Thus one rephrases (27.72) as the "generalized Friedmann equation,"
acts as an "effective potential" for the dynamics of the expansion. The constant term K_(o)K_{o} represents one sixth of the intrinsic curvature of the model universe today. Its negative, -K_(o)-K_{o}, plays the role of an "effective energy" in the generalized Friedmann equation (Box 27.5). All the qualitative features of the cosmology can be read off from the curve for the effective potential as a function of (a//a_(o))\left(a / a_{o}\right) and from the value of K_(o)K_{o}.
For a quantitative analysis, the log-log diagram of Figure 27.5 is often more useful than the straight linear plot of VV against (a//a_(0))\left(a / a_{0}\right) of Box 27.5.
All the limiting features shown in the varied types of cosmology have been encountered before in the analysis of the elementary Friedmann cosmology (big bang out of a configuration of infinite compaction; reaching a maximum expansion at a turning point, or continued expansion to a Minkowski universe; recollapse to
infinite density) or lend themselves to simple visualization (static but unstable Einstein universe; "hesitation" model; "turnaround" model), except for the even more rapid expansion that occurs when Lambda\Lambda is positive and the dimension aa has surpassed a certain critical value. In this expansion, aa eventually increases as exp[(Lambda//3)^(1//2)t]\exp \left[(\Lambda / 3)^{1 / 2} t\right] irrespective of the openness or closure of the universe (k=0,+-1)(k=0, \pm 1). This expansion dominates every other feature of the cosmology. Therefore, in discussing it, it is appropriate to suppress every other feature of the cosmology, take the density of matter to be negligible, and take k=0k=0 (hypersurfaces of homogeneity endowed with flat 3-space geometry). In this limit, one has the following empty-space solution of Einstein's field equation with cosmological constant:
{:(27.76)ds^(2)=-dt^(2)+a_(o)^(2)e^(2(Lambda//3)^(1//2)t)[dchi^(2)+chi^(2)(dtheta^(2)+sin^(2)theta dphi^(2))].:}\begin{equation*}
d s^{2}=-d t^{2}+a_{o}^{2} e^{2(\Lambda / 3)^{1 / 2} t}\left[d \chi^{2}+\chi^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] . \tag{27.76}
\end{equation*}
This "de Sitter universe" [de Sitter (1917a,b)] may be regarded as a four-dimensional surface,
Because of its beautiful group-theoretical properties and invariance with respect to 5xx4//2=105 \times 4 / 2=10 independent rotations, the de Sitter geometry has been the subject of scores of mathematical investigations. The physical implications of a cosmology following the de Sitter model are described for example by Robertson and Noonan (1968, especially their §16.2\S 16.2 ). The de Sitter model is the only model obeying Einstein's equations (with Lambda!=0\Lambda \neq 0 ) which (1) continually expands and (2) looks the same to any observer who moves with the cosmological fluid, regardless of his position or his time. Any model of the universe satisfying condition (2) is said to obey the so-called "perfect cosmological principle." This phrase arose in the past in studying models that lie outside the framework of general relativity, models in which matter is envisaged as continuously being created, and to which the name of "steady-state universe" has been given. Any such model has been abandoned by most investigators today, not least because it gives no satisfactory account of the 2.7 K background radiation.
Other non-Einsteinian cosmologies:
(1) steady-state model
Box 27.5 EFFECT OF VALUE OF COSMOLOGICAL CONSTANT AND OF INTRINSIC CURVATURE OF MODEL UNIVERSE "TODAY" ON THE PREDICTED COURSE OF COSMOLOGY
The "effective potential" VV in the generalized Friedmann equation (27.74) is represented schematically here as a function of the expansion ratio a//a_(0)a / a_{0}. The diagram illustrates the influence on the cosmology of (1) the cosmological constant Lambda\Lambda (determines the behavior of the effective potential at large values of a//a_(o)a / a_{o}; see dashed curves) and (2) the value adopted for K_(o)=K_{o}= (one sixth of the intrinsic curvature of 3 -space at the present epoch). The value of the quantity -K_(o)-K_{o} determines the "effective energy level" and is shown in the diagram as a horizontal line. The difference between this horizontal line and the effective potential determines (a_(o)^(-1)da//dt)^(2)\left(a_{o}^{-1} d a / d t\right)^{2}. Regions where this difference is negative are inaccessible. From the diagram one can read off the histories of 3 -space on the facing page.
The diagram is schematic, not quantitative. Representative values might be Lambda_("conv ")=0\Lambda_{\text {conv }}=0 or +-3xx10^(-28)g//cm^(3);quadrho_(mo," conv ")=10^(-30)g//cm^(3)\pm 3 \times 10^{-28} \mathrm{~g} / \mathrm{cm}^{3} ; \quad \rho_{m o, \text { conv }}=10^{-30} \mathrm{~g} / \mathrm{cm}^{3} or rho_(mo," conv ")=10^(-28)g//cm^(3);\rho_{m o, \text { conv }}=10^{-28} \mathrm{~g} / \mathrm{cm}^{3} ; and (a_(o)^(-1)da//dt)^(2)=H_(o)^(2)=(1//20 xx10^(9)yr)^(2)\left(a_{o}^{-1} d a / d t\right)^{2}=H_{o}^{2}=\left(1 / 20 \times 10^{9} \mathrm{yr}\right)^{2} or 3.8 xx10^(-29)g//cm^(3)3.8 \times 10^{-29} \mathrm{~g} / \mathrm{cm}^{3}. At small values of a//a_(o)a / a_{o} the cosmological term -(Lambda//3)(a//a_(o))^(2)-(\Lambda / 3)\left(a / a_{o}\right)^{2} is negligible. Not negligible at small values of (a//a_(0))\left(a / a_{0}\right) is the difference between a model universe curved only by the density of matter (the dashed curve in the diagram) and one curved also by a density of radiation (the full curve). The different dependence of "radius" and density on time at early times in these two cases of a matter-dominated cosmology and a radiationdominated cosmology is spelled out in the last part of Box 27.3, giving in the one case rho=1//6pit^(2)\rho=1 / 6 \pi t^{2} and in the other rho=3//32 pit^(2)\rho=3 / 32 \pi t^{2}.
Intrinsic curvature
of space today
Intrinsic curvature
of space today| Intrinsic curvature |
| :--- |
| of space today |
Universe starts in a condition of infinite density. It ex-
pands to a maximum extent (or minimum density) governed
by the value of Lambda\Lambda. It then recontracts at an ever increasing
rate to a condition of infinite density.
Universe starts in a condition of infinite density. It ex-
pands to a maximum extent (or minimum density) governed
by the value of Lambda. It then recontracts at an ever increasing
rate to a condition of infinite density.| Universe starts in a condition of infinite density. It ex- |
| :--- |
| pands to a maximum extent (or minimum density) governed |
| by the value of $\Lambda$. It then recontracts at an ever increasing |
| rate to a condition of infinite density. |
Universe starts in a condition of infinite density. It ex-
pands. Ultimately the rate of expansion reaches a steady
rate, da/dt =1=1. The 4-geometry is Minkowski flat spacetime.
Universe starts in a condition of infinite density. It ex-
pands. Ultimately the rate of expansion reaches a steady
rate, da/dt =1. The 4-geometry is Minkowski flat spacetime.| Universe starts in a condition of infinite density. It ex- |
| :--- |
| pands. Ultimately the rate of expansion reaches a steady |
| rate, da/dt $=1$. The 4-geometry is Minkowski flat spacetime. |
Only the curvature of the spacelike slices taken through this
flat 4-geometry gives the 3-geometry its hyperbolic character
[see equation (27.70)].
"Intrinsic curvature
of space today" negative "Cosmology"
"Hyperbolic;
K_(o) negative" "Universe starts in a condition of infinite density. It ex-
pands to a maximum extent (or minimum density) governed
by the value of Lambda. It then recontracts at an ever increasing
rate to a condition of infinite density."
"Hyperbolic;
K_(o) negative" "Universe starts in a condition of infinite density. It ex-
pands. Ultimately the rate of expansion reaches a steady
rate, da/dt =1. The 4-geometry is Minkowski flat spacetime."
Only the curvature of the spacelike slices taken through this
flat 4-geometry gives the 3-geometry its hyperbolic character
[see equation (27.70)]. | Intrinsic curvature <br> of space today | negative | Cosmology |
| :--- | :--- | :--- |
| Hyperbolic; <br> $K_{o}$ negative | Universe starts in a condition of infinite density. It ex- <br> pands to a maximum extent (or minimum density) governed <br> by the value of $\Lambda$. It then recontracts at an ever increasing <br> rate to a condition of infinite density. | |
| Hyperbolic; <br> $K_{o}$ negative | Universe starts in a condition of infinite density. It ex- <br> pands. Ultimately the rate of expansion reaches a steady <br> rate, da/dt $=1$. The 4-geometry is Minkowski flat spacetime. | |
| Only the curvature of the spacelike slices taken through this | | |
| flat 4-geometry gives the 3-geometry its hyperbolic character | | |
| [see equation (27.70)]. | | |
Closed; K_(o)K_{o} positive
A positive and exactly equal to the critical value, Lambda=Lambda_("crit ")\Lambda=\Lambda_{\text {crit }}, that puts the "summit of the potential" into coincidence with -K_(o)-K_{o}
Situation similar to that of a pencil with its tip dug into the table and provided with just enough energy to rise asymptotically in infinite time to the vertical position. Universe starts from a compact configuration and expanding approaches a certain radius ("Einstein radius", a_(E)a_{E} ) according to a law of the form
Or (Einstein's original proposal, when he thought that the universe is static, and added the "cosmological term," against his will, to the field equation to permit a static universe) the representative point sits forever at the "summit of the effective potential" (Einstein universe). Aside from contradicting present-day evidence on expansion, this configuration has the same instability as does a pencil trying to stand on its tip. The least disturbance will cause it to "fall" either way, toward collapse or toward accelerating expansion, in the expansion case ultimately approaching the law
Closed; K_(o)K_{o} positive Lambda\Lambda less positive than the critical value: 0 < Lambda < Lambda_("erit ")(K_(o))0<\Lambda<\Lambda_{\text {erit }}\left(K_{o}\right)
Motion on the large aa side of the "potential barrier." Far back in the past the model universe has enormous dimensions, but is also contracting with enormous rapidity, in approximate accord with the formula
Figure 27.5. (facing page)
Log-log plot of the effective potential V(a)V(a) of equation (27.75) and Box 27.5 as it enters the generalized Friedmann equation
Horizontally is given the expansion ratio referred to (a//a_(o))_("today ")=1\left(a / a_{o}\right)_{\text {today }}=1 as standard of reference. Vertically is given the value of V(a//a_(o))V\left(a / a_{o}\right) in the geometric units of cm^(-2)\mathrm{cm}^{-2}. The supplementary scale at the right translates to -(c^(2)//G)(3//8pi)V(a//a_(0))-\left(c^{2} / G\right)(3 / 8 \pi) V\left(a / a_{0}\right) as an equivalent density, expressed in g//cm^(3)\mathrm{g} / \mathrm{cm}^{3}. The contribution of radiation density to the effective potential is indicated by the wavy line in the diagram. It is normalized to a value of the radiation density today of rho_("ro ")=10^(-33)g//cm^(3)\rho_{\text {ro }}=10^{-33} \mathrm{~g} / \mathrm{cm}^{3} and has a logarithmic slope of two. The contribution of matter density to the "effective potential" has a logarithmic slope of unity. Two choices are made for it, corresponding to a density of matter today of rho_(mo)=10^(-30)g//cm^(3)\rho_{m o}=10^{-30} \mathrm{~g} / \mathrm{cm}^{3} and rho_(mo)=10^(-28)g//cm^(3)\rho_{m o}=10^{-28} \mathrm{~g} / \mathrm{cm}^{3} (dashed lines in the diagram). The total effective potential in the two cases is also indicated in the diagram: a heavy line for the case rho_(mo)=10^(-30)g//cm^(3)\rho_{m o}=10^{-30} \mathrm{~g} / \mathrm{cm}^{3} (no cosmological term included) and a light line for the case rho_(mo)=10^(-28)g//cm^(3)\rho_{m o}=10^{-28} \mathrm{~g} / \mathrm{cm}^{3}. In this second case, a cosmological term is included, with the cosmological constant given by (3//8pi)(Lambda//3)=10^(-29)g//cm^(3)(3 / 8 \pi)(\Lambda / 3)=10^{-29} \mathrm{~g} / \mathrm{cm}^{3}. The line describing the contribution of this term has a negative slope of magnitude two (dashed line). The horizontal or "level line" is drawn for a value of the Hubble expansion rate today, H_(o)H_{o}, equal to 1//(20 xx10^(9):}1 /\left(20 \times 10^{9}\right. years )). The vertical separation on the log\log plot between the potential curve and the level line gives the ratio -V//H_(o)^(2)-V / H_{o}^{2}. This ratio as evaluated at any time tt has the value a^(˙)^(2)(t)//a^(˙)_(o)^(2)+K_(o)H_(o)^(-2)\dot{a}^{2}(t) / \dot{a}_{o}^{2}+K_{o} H_{o}^{-2}, where a^(˙)-=da//dt\dot{a} \equiv d a / d t. As evaluated "today" (a//a_(o)=1)\left(a / a_{o}=1\right) this ratio has the value 1+K_(o)H_(o)^(-2)1+K_{o} H_{o}^{-2}. Knowing the Hubble expansion rate H_(o)^(2)H_{o}^{2} today, and knowing (or trying a certain set of parameters for) the potential curve, one can therefore deduce from the spread between the two the value of 1+K_(o)H_(o)^(-2)1+K_{o} H_{o}^{-2}, hence the value of K_(o)H_(o)^(-2)K_{o} H_{o}^{-2}, hence the present value, K_(o)K_{o}, of the curvature factor. As an example, consider the case of the low-density universe (heavy line) and read off "today's" value, 1+K_(o)H_(o)^(-2)=0.2231+K_{o} H_{o}^{-2}=0.223. From this follows K_(o)=-0.777H_(o)^(2)K_{o}=-0.777 H_{o}^{2} (open or hyperbolic universe), hence k=-1k=-1 and a_(o)=(k//K_(0))^(1//2)=(1//0.777)^(1//2)20 xx10^(9)yr=22.7 xx10^(9)yra_{o}=\left(k / K_{0}\right)^{1 / 2}=(1 / 0.777)^{1 / 2} 20 \times 10^{9} \mathrm{yr}=22.7 \times 10^{9} \mathrm{yr}. For the high-density model universe, with rho_(mo)=10^(-28)g//cm^(3)\rho_{m o}=10^{-28} \mathrm{~g} / \mathrm{cm}^{3}, one similarly finds 1+K_(o)H_(o)^(-2)=24.51+K_{o} H_{o}^{-2}=24.5, hence K_(o)=+23.5H_(o)^(2)K_{o}=+23.5 H_{o}^{2}, hence k=+1k=+1 (closed universe) and a_(o)=(k//K_(o))^(1//2)=(1//23.5)^(1//2)20 xx10^(9)yr=4.12 xx10^(9)yra_{o}=\left(k / K_{o}\right)^{1 / 2}=(1 / 23.5)^{1 / 2} 20 \times 10^{9} \mathrm{yr}=4.12 \times 10^{9} \mathrm{yr}. Expansion stops, if and when it stops, at that stage when the ratio -V//H_(o)^(2)-V / H_{o}^{2} between the potential and the level line, or a^(˙)^(2)(t)//a^(˙)_(o)^(2)+K_(o)H_(o)^(-2)\dot{a}^{2}(t) / \dot{a}_{o}^{2}+K_{o} H_{o}^{-2}, falls from its "present value" of 1+K_(o)H_(o)^(-2)1+K_{o} H_{o}^{-2} to 0+K_(o)H_(o)^(-2)0+K_{o} H_{o}^{-2}; that is, from 0.223 to -0.777 in the one case, and from 24.5 to 23.5 in the other case. This log-log\log -\log plot should be replaced by the linear plot of Box 27.5 when Lambda < 0\Lambda<0.
(2) hierarchic model
However great a departure it is from Einstein's concept of cosmology to give any heed to a cosmological constant or an open universe, it is a still greater departure to contemplate a "hierarchic model" of the universe, in which clusters of galaxies, and clusters of clusters of galaxies, in this part of the universe are envisaged to grade off in density with distance, with space at great distances becoming asymptotically flat [Alfvén and Klein (1962), Alfvén (1971), Klein (1971), Moritz (1969), de Vaucouleurs (1971), Steigman (1971)]. The viewpoint adopted here is expressed by Oskar Klein in these words, "Einstein's cosmology was adapted to the discovery by Hubble that the observed part is expanding; the so-called cosmological postulate has been used as a kind of an axiomatic background which, when analyzed, makes it appear that this expansion is shared by a very big, but still bounded system. This implies that our expanding metagalaxy is probably just one of a type of stellar objects in different phases of evolution, some expanding and some contracting."
The contrast between the hierarchic cosmology and Einstein's cosmology [Einstein (1931) advocates a closed Friedmann cosmology] appears nowhere more strongly than here, that the one regards asymptotically flat spacetime as a requirement; the other, as an absurdity. "Only the genius of Riemann, solitary and uncomprehended," Einstein (1934) puts it, "had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible." That statement epitomizes cosmology today.
But today's view of cosmology, as dominated by Einstein's boundary condition of closure (k=+1)(k=+1) and his belief in Lambda=0\Lambda=0, need not be accepted on faith forever. Einstein's predictions are clear and definite. They expose themselves to destruction. Observational cosmology will ultimately confirm or destroy them, as decisively as it has already destroyed the 1920 belief in a static universe and the 1948 steady-state models (see Box 27.7 on the history of cosmology).
EXERCISES
Exercise 27.11. ON SEEING THE BACK OF ONE'S HEAD
Can a being at rest relative to the "cosmological fluid" ever see the back of his head by means of photons that travel all the way around a closed model universe that obeys the Friedmann cosmology and has a non-zero cosmological constant (see the entries in Box 27.3 for the case of a zero cosmological constant)?
Exercise 27.12. DO THE CONSERVATION LAWS FORBID THE PRODUCTION OF PARTICLE-ANTIPARTICLE PAIRS OUT OF EMPTY SPACE BY TIDAL GRAVITATION FORCES?
Find out what is wrong with the following argument: "The classical equations
are not compatible with the production of pairs, since they lead to the identity T_(alpha)^(beta);beta-=0T_{\alpha}{ }^{\beta} ; \beta \equiv 0. Let the initial state be vacuum, and let T_(alpha beta)T_{\alpha \beta} and its derivative be equal to zero on the hypersurface t=t= const or t=-oot=-\infty. It then follows from T_(alpha)^(beta)_(;beta)=0T_{\alpha}{ }^{\beta}{ }_{; \beta}=0, that the vacuum is always conserved." [Answer: See Zel'dovich (1970, 1971, 1972). Also see §30.8.]
Exercise 27.13. TURN-AROUND UNIVERSE MODEL NEGLECTING MATTER DENSITY
If turn-around (minimum radius) occurs far to the right (large aa ) of the maximum of the potential V(a)V(a) in equation (27.75), the matter terms will be negligible. Let rho_(mo)=rho_(ro)=0\rho_{m o}=\rho_{r o}=0. Then (what signs of k,Lambdak, \Lambda are needed for turn-around?), solve to show that Lambda=3(a_("min "))^(-2)\Lambda=3\left(a_{\text {min }}\right)^{-2}, H=(a_("min "))^(-1)tanh(t//a_(min))H=\left(a_{\text {min }}\right)^{-1} \tanh \left(t / a_{\min }\right) near turnaround (t=0)(t=0) and that the deceleration parameter q-=-(1//H^(2)a)(d^(2)a//dt^(2))q \equiv-\left(1 / H^{2} a\right)\left(d^{2} a / d t^{2}\right) has the value
Neglect radiation in equation (27.75) but assume K_(o)K_{o} and Lambda\Lambda to be chosen so that the universe spent a very long time with a(t)a(t) near a_(h)(a_(h):}a_{h}\left(a_{h}\right. measures location of highest point of the barrier, or the size of the universe at which the universe is most sluggish). Choose a_(h)=a_(o)//3a_{h}=a_{o} / 3 to produce an abnormally great number of quasar redshifts near z=2z=2 [as Burbidge and Burbidge (1969a,b) believe to be the case, though their opinion is not shared by all observers]. Show (a) that the density of matter now would account for only 10 per cent of the value of H_(o)^(2)=(a^(˙)//a)_("now ")^(2)H_{o}^{2}=(\dot{a} / a)_{\text {now }}^{2} in equation (27.75) ["missing matter", i.e., K_(o)K_{o} and Lambda\Lambda terms, account for 90 per cent], (b) that a_(h)≃20^(1//2)H_(o)^(-1)a_{h} \simeq 20^{1 / 2} H_{o}^{-1}, and (c) that the deceleration parameter defined in the previous exercise, as evaluated "today", has the value q_(o)=-13//10q_{o}=-13 / 10.
Exercise 27.15. UNIVERSE OPAQUE TO BLACK-BODY RADIATION AT A NONSINGULAR PAST TURN-AROUND REQUIRES IMPOSSIBLE PARAMETERS
From a plot like that in Box 27.5, construct a model of the universe that contains 2.7 K black-body radiation at the present, but, with k=+1k=+1 and Lambda > 0\Lambda>0, had a past turn-around (minimum radius) at which the blackbody temperature reached 3,000K3,000 \mathrm{~K} where hydrogen would be ionized. Try to use values of H_(o)^(-1)H_{o}^{-1} and rho_(mo)\rho_{m o} that are as little as possible smaller than presently accepted values.
Box 27.6 ALEXANDER ALEXANDROVITCH FRIEDMANN
St. Petersburg, June 17, 1888-Leningrad, September 15, 1925
Graduated from St. Petersburg University, 1909; doctorate, 1922; 1910, mathematical assistant in the Institute of Bridges and Roads; 1912, lecturer on differential calculus in the Institute of Mines; 1913, physicist in the Aerological Institute of Pavlov; dirigible ascent in preparation for observing eclipse of the sun of August 1914; volunteer in air corps on war front near Osovets, 1914; head of military air navigation service, 1916-1917; professor of mechanics at Perm University, 1918; St. Petersburg University, 1920; lectures in hydrodynamics, tensor analysis; author of books, Experiments in the Hydromechanics of Compressible Liquids and The World as Space and Time, and the path-breaking paper, On the Curvature of Space, 1922; a director of researches in the department of theoretical meteorology of the Main Geophysical Laboratory, Leningrad, and, from February 1925 until his death of typhoid fever seven months later, director of that Laboratory; with L. V. Keller "introduced the concept of coupling moments, i.e., mathematical expectation values for the products of pulsations of hydrodynamic elements at different points and at different instants . . . to elucidate the physical structure of turbulence" [condensed from Polubarinova-Kochina (1963), which also contains a bibliography of items by and about Friedmann].
Box 27.7 SOME STEPS IN COSMOLOGY ON THE WAY TO WIDER PERSPECTIVES AND FIRMER FOUNDATIONS [For general reference on the history of cosmology, see among others Munitz (1957), Nasr (1964), North (1965), Peebles (1971), Rindler (1969), and Sciama (1971); and especially see Peebles and Sciama for bibliographical references to modern developments listed below in abbreviated form.]
A. Before the Twentieth Century
Concepts of very early Indian resmology [summarized by Zimmer (1946)]: "One thousand mahāyugas- 4,320,000,0004,320,000,000 years of human reckoning-constitute a single day of Brahmā, a single kalpa. . . . I have known the dreadful dissolution of the universe. I have seen all perish, again and again, at every cycle. At that terrible time, every single atom dissolves into the primal, pure waters of eternity, whence all originally arose."
Plato, ca. 428 to ca. 348 b.c. [from the Timaeus, written late in his life, as translated by Cornford (1937)]: "The world [universe] has been fashioned on the model of that which is comprehensible by rational discourse and understanding and is always in the same state. . . . this world came to be . . . a living creature with soul and reason. . . . its maker did not make two worlds nor yet an indefinite number; but this Heaven has come to be and is and shall be hereafter one and unique. . . . he fashioned it complete and free from age and sickness. . . . he turned its shape rounded and spherical . . . It had no need of eyes, for nothing visible was left outside; nor of hearing, for there was nothing outside to be heard. . . . in order that Time might be brought into being, Sun and Moon and five other stars-'wanderers,' as they are called-were made to define and preserve the numbers of Time. . . . the generation of this universe was a mixed result of the combination of Necessity and Reason . . . we must also bring in the Errant Cause. . . . that which is to receive in itself all kinds [all forms] must be free from all characters [all form] . . . . For this reason, then, the mother and Receptacle of what has come to be visible and otherwise sensible must not be called earth or air or fire or water . . . but a nature invisible and characterless, all-receiving, partaking in some very puzzling way of the intelligible, and very hard to apprehend."
Aristotle, 384-322 в.c. [from On the Heavens, as translated by Guthrie (1939)]: "Throughout all past time, according to the records handed down from generation to generation, we find no trace of change either in the whole of the outermost heaven or in any one of its proper parts. . . . the shape of the heaven must be spherical. . . . From these considerations [motion invariably in a straight line toward the center; regularity of rising and setting of stars; natural motion of earth toward the center of the universe] it is clear that the earth does not move, neither does it lie anywhere but at the center. . . . the earth . . . must have grown in the form of a sphere. This [shape of segments cut out of moon at time of eclipse of moon; and ability to see
in Egypt stars not visible in more northerly lands] proves both that the earth is spherical and that its periphery is not large . . . Mathematicians who try to calculate the circumference put it at 400,000 stades [1[1 stade =600=600 Greek feet =606=606 English feet; thus 24.24 xx10^(7)ft//(6080.2ft//24.24 \times 10^{7} \mathrm{ft} /(6080.2 \mathrm{ft} / nautical mile )=39,900)=39,900 nautical miles-the oldest recorded calculation of the earth's circumference, and reportedly known to Columbus- 85 per cent more than the true circumference, 60 xx360=21,60060 \times 360=21,600 nautical miles]."
Aristotle [from the Metaphysics, as translated by Warrington (1956)]: "Euxodus [of Cnidos, 408-355 b.c.] supposed that the motion of the sun and moon involves, in each case, three spheres. . . He further assumed that the motion of the planets involves, in each case, four spheres. . . . Calippus [of Cyzicus, flourished 330 b.c.] . . . considered that, in the light of observation, two more spheres should be added to the sun, two to the moon, and one more to each of the other planets."
Eratosthenes, 276-194 b.c. [a calculation attributed to him by Claudius Ptolemy, who observed at Alexandria from 127 to 141 or 151 A.D., in his Almagest, I, § 12 ; see the translation by Taliaferro (1952)]: (( Maximum distance of moon from earth )=(64(1)/(6)))=\left(64 \frac{1}{6}\right) (radius of earth )); (( Minimum distance of sun from earth )=(1,160))=(1,160) (radius of earth).
Abū 'Alī al-Husain ibn 'Abdallāh ibn Sīnā, otherwise known as Avicenna, 980-1037; physician, philosopher, codifier of Aristotle, and one of the most influential of those who preserved Greek learning and thereby made possible its transmission to mediaeval Europe [quoted in Nasr (1964), p. 225]: "Time is the measure of motion."
From the Rasä vec(il)\overrightarrow{i l}, a 51 -treatise encyclopedia, sometimes known as the Koran after the Koran, of the Ikhwān al-Safā' or Brothers of Sincerity, whose main center was at Basra, Iraq, roughly A.D. 950-1000950-1000 [as quoted by Nasr (1964), p. 64; see p. 78 for a list of distances to the planets (in terms of Earth radii) taken from the Rasä̉il, as well as sizes of planets and the motions of rotation of the various Ptolemaic carrier-spheres]: Space is "a form abstracted from matter and existing only in the consciousness."
Abū Raihān al-Bīrūnī, 973-1030, a scholar, but concerned also with experiment, observation, and measurement, who calculated the circumference of the Earth from measurements he made in India as 80,780,03980,780,039 cubits (about 4 per cent larger than the value accepted today), and gave a table of distances to the planets [as quoted in Nasr (1964), pp. 120 and 130]: "Both [kinds of eclipses] do not happen together except at the time of the total collapse of the universe."
Étiene Tempier, Bishop of Paris, in 1277, to settle a controversy then dividing much of the French theological community, ruled that one could not deny the power of God to create as many universes as He pleases.
Roger Bacon, 1214-1294, in his Opus Majus (1268), gave the diameter of the sphere that carries the stars, on the authority of Alfargani, as 130,715,000130,715,000 Roman miles
Box 27.7 (continued)
[mile equal to 1,000 settings down of the right foot]; the volume of the sun, 170 times that of the Earth; first-magnitude star, 107 times; sixth-magnitude, 18 times Earth.
Nicolas Cusanus, 1401-1464 [from Of Learned Ignorance (1440), as translated by Heron (1954)]: "Necessarily all parts of the heavens are in movement. . . . It is evident from the foregoing that the Earth is in movement . . . the world [universe], its movement and form . . . will appear as a wheel in a wheel, a sphere in a sphere without a center or circumference anywhere. . . . It is now evident that this Earth really moves, though to us it seems stationary. In fact, it is only by reference to something fixed that we detect the movement of anything. How would a person know that a ship was in movement, if . . . the banks were invisible to him and he was ignorant of the fact that water flows?"
Nicolaus Copernicus, February 19, 1473, to May 24, 1543 [from De Revolutionibus Orbium Coelestrum (1543), as translated by Dobson and Brodetsky (1947)]: "I was induced to think of a method of computing the motions of the spheres by nothing less than the knowledge that the mathematicians are inconsistent in these investigations. . . . they cannot explain or observe the constant length of the seasonal year. . . . some use only concentric circles, while others eccentrics and epicycles. . . . Nor have they been able thereby to discern or deduce the principal thing-namely the shape of the universe and the unchangeable symmetry of its parts. . . .
"I found first in Cicero that Nicetas had realized that the Earth moved. Afterwards I found in Plutarch [ ∼\sim A.D. 46-120]dots46-120] \ldots. . The rest hold the Earth to be stationary, but Philolaus the Pythagorean [born ∼480\sim 480 B.c.] says that she moves around the (central) fire on an oblique circle like the Sun and Moon. Heraclides of Pontus [flourished in 4th century b.c.] and Ecphantus the Pythagorean also make the Earth to move, not indeed through space but by rotating round her own center as a wheel on an axle from West to East.' Taking advantage of this I too began to think of the mobility of the Earth. . . .
"Should we not be more surprised if the vast Universe revolved in twenty-four hours, than that little Earth should do so? . . . Idle therefore is the fear of Ptolemy that Earth and all thereon would be disintegrated by a natural rotation. . . . That the Earth is not the center of all revolutions is proved by the apparently irregular motions of the planets and the variations in their distances from the Earth. . . . We therefore assert that the center of the Earth, carrying the Moon's path, passes in a great orbit among the other planets in an annual revolution round the Sun; that near the Sun is the center of the Universe; and that whereas the Sun is at rest, any apparent motion of the Sun can be better explained by motion of the Earth. . . . Particularly Mars, when he shines all night, appears to rival Jupiter in magnitude, being distinguishable only by his ruddy color; otherwise he is scarce equal to a star of the second magnitude, and can be recognized only when his movements are
carefully followed. All these phenomena proceed from the same cause, namely Earth's motion. . . . That there are no such phenomena for the fixed stars proves their immeasurable distance, compared to which even the size of the Earth's orbit is negligible and the parallactic effect unnoticeable."
Thomas Digges, 1546-1595 [in a Perfit Description of the Caelestiall Orbes according to the most aunciente doctrine of the Pythagoreans, latelye reuiued by Copernicus and by Geometricall Demonstrations approued (1576), the principal vehicle by which Copernicus reached England, as quoted in Johnson (1937)]: "Of whiche lightes Celestiall it is to bee thoughte that we onely behoulde sutch as are in the inferioure partes of the same Orbe, and as they are hygher, so seeme they of lesse and lesser quantity, euen tyll our sighte beinge not able farder to reach or conceyue, the greatest part rest by reason of their wonderfull distance inuisible vnto vs."
Giordano Bruno, born ca. 1548, burned at the stake in the Campo dei Fiori in Rome, February 17, 1600 [from On the Infinite Universe and Worlds, written on a visit to England in 1583-1585, as translated by Singer (1950)]: "Thus let this surface be what it will, I must always put the question, what is beyond? If the reply is NOTHING, then I call that the VOID or emptiness. And such a Void or Emptiness hath no measure and no outer limit, though it hath an inner; and this is harder to imagine than is an infinite or immense universe. . . . There are then innumerable suns, and an infinite number of earths revolve around those suns, just as the seven we can observe revolve around this sun which is close to us."
Johann Kepler established the laws of elliptic orbits and of equal areas (1609), and established the connection between planetary periods and semimajor axes (1619).
Galileo Galilei observed the satellites of Jupiter and realized they provided support for Copernican theory, and interpreted the Milky Way as a collection of stars (1610). In 1638 he wrote:
"Salvati. Now what shall we do, Simplicio, with the fixed stars? Do we want to sprinkle them through the immense abyss of the universe, at various distances from any predetermined point, or place them on a spherical surface extending around a center of their own so that each of them will be at the same distance from that center?
"Simplicio. I had rather take a middle course, and assign to them an orb described around a definite center and included between two spherical surfaces . . ."
Isaac Newton (1687): "Gravitation toward the sun is made up out of the gravitations toward the several particles of which the body of the sun is composed, and in receding from the sun decreases accurately as the inverse square of the distances as far as the orbit of Saturn, as evidently appears from the quiescence of the aphelion of the planets."
Isaac Newton [in a letter of Dec. 10, 1692, to Richard Bentley, quoted in Munitz (1957)]: "If the matter of our sun and planets and all the matter of the universe were evenly scattered throughout all the heavens, and every particle had an innate gravity toward all the rest, and the whole space throughout which this matter was
Box 27.7 (continued)
scattered was but finite, the matter on the outside of this space would, by its gravity, tend toward all the matter on the inside and, by consequence, fall down into the middle of the whole space and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space, it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses scattered at great distances from one to another throughout all that infinite space. And thus might the sun and fixed stars be formed."
Christiaan Huygens, 1629-1695 [in his posthumously published Cosmotheoros (1698)]: "Seeing then that the stars . . . are so many suns, if we do but suppose one of them [Sirius, the Dog-star] equal to ours, it will follow [details, including telescope directed at sun; thin plate; hole in it; comparison with Sirius] . . . that his distance to the distance of the sun from us is as 27,664 to 1 . . . Indeed it seems to me certain that the universe is infinitely extended."
Edmund Halley (1720): "If the number of the Fixt Stars were more than finite, the whole superficies of their apparent Sphere [i.e., the sky] would be luminous" [by today's reasoning the same temperature as the surface of the average star; this is known today as Olber's paradox, or the paradox of P. L. de Chéseaux (1744) and Heinrich Wilhelm Matthias Olbers (1826)].
Thomas Wright of Durham (1750): "To . . . solve the Phaenomena of the Via Lactea . . . granted . . . that the Milky Way is formed of an infinite number of small Stars . . . imagine a vast infinite gulph, or medium, every way extended like a plane, and inclosed between two surfaces, nearly even on both sides. . . . Now in this space let us imagine all the Stars scattered promiscuously, but at such an adjusted distance from one another, as to fill up the whole medium with a kind of regular irregularity of objects. [Considering its appearance] "to an eye situated . . . anywhere about the middle plane" . . . all the irregularity we observe in it at the Earth, I judge to be entirely owing to our Sun's position . . . and the diversity of motion . . . amongst the stars themselves, which may here and there . . . occasion a cloudy knot of stars."
Immanuel Kant, 1724-1804 (1755): "It was reserved for an Englishman, Mr. Wright of Durham, to make a happy step . . . we will try to discover the cause that has made the positions of the fixed stars come to be in relation to a common plane. . . . granted . . . that the whole host of [the fixed stars] are striving to approach each other through their mutual attraction . . ruin is prevented by the action of the centrifugal forces . . . the same cause [centrifugal force] . . . has also so directed their orbits that they are all related to one plane. . . . [The needed motion is calculated to be] one degree [or less] in four thousand years; . . . careful observers . . . will be required for it. . . . Mr. Bradley has observed almost imperceptible displacements of the stars" [known from later work to be caused by aberration (effect of observer velocity) rather than real parallax (effect of position of observer)].
Asks for the first time how a very remote galaxy would appear: "circular if its plane is presented directly to the eye, and elliptical if it is seen from the side or obliquely. The feebleness of its light, its figure, and the apparent size of its diameter will clearly distinguish such a phenomenon when it is presented, from all the stars that are seen single. . . this phenomenon . . . has been distinctly perceived by different observers [who] . . . have been astonished at its strangeness. . . . Analogy thus does not leave us to doubt that these systems [planets, stars, galaxies] have been formed and produced . . . out of the smallest particles of the elementary matter that filled empty space."
Goes on to consider seriously "the successive expansion of the creation [of planets, stars, galaxies] through the infinite regions of space that have the matter for it. . . . attraction is just that universal relation which unites all the parts of nature in one space. It reaches, therefore, to . . . all the distance of nature's infinitude."
Johann Heinrich Lambert, 1728-1777 (1761): "The fixed stars obeying central forces move in orbits. The Milky Way comprehends several systems of fixed stars. . . . Each system has its center, and several systems taken together have a common center. Assemblages of their assemblages likewise have theirs. In fine, there is a universal center for the whole world round which all things revolve." [First spelling out of a "hierarchical model" for the universe, later taken up by C. V. I. Charlier and by H. Alfvén and O. Klein (1962); see also O. Klein (1966 and 1971)].
Auguste Comte (1835) concluded that it is meaningless to speak of the chemical composition of distant stars because man will never be able to explore them; "the field of positive philosophy lies wholly within the limits of our solar system, the study of the universe being inaccessible in any positive sense."
The first successful determination of the parallax [1 second of parallax: 1pc=1 \mathrm{pc}=3.08 xx10^(18)cm=3.26lyr3.08 \times 10^{18} \mathrm{~cm}=3.26 \mathrm{lyr} ] of any star was made in 1838 (for alpha\alpha Centauri by Henderson, for alpha\alpha Lyrae by Struve, and for 61 Cygni by Bessel).
B. The Twentieth Century
Derivation by James Jeans in 1902 of the critical wavelength that separates shortwavelength acoustical modes of vibration of a hot primordial gas and longer wavelength modes of commencement of gravitational condensation of this gas. Application of these considerations by P. J. E. Peebles and R. H. Dicke in 1968 to explain why globular star clusters have masses of the order of 10^(5)M_(o.)10^{5} \mathrm{M}_{\odot}.
Investigations of cosmic rays from first observation by V. F. Hess and W. Kolhörster in 1911-1913 to date; determination that the energy density in interstellar space (in this galaxy) is about 1eV//cm^(3)1 \mathrm{eV} / \mathrm{cm}^{3} or 10^(-12)erg//cm^(3)10^{-12} \mathrm{erg} / \mathrm{cm}^{3}, comparable to the density of energy of starlight, to the kinetic energy of clouds of ionized interstellar gas, averaged over the galaxy, and to the energy density of the interstellar
Box 27.7 (continued)
magnetic field ( ∼10^(-5)\sim 10^{-5} gauss). In connection with this equality, see especially E. Fermi (1949).
Discovery by Henrietta Leavitt in 1912 that there is a well-defined relation between the period of a Cepheid variable and its absolute luminosity.
First determination of the radial velocity of a galaxy by V. M. Slipher in 1912: Andromeda approaching at 200km//sec200 \mathrm{~km} / \mathrm{sec}. Thirteen galaxies investigated by him by 1915; all but two receding at roughly 300km//sec300 \mathrm{~km} / \mathrm{sec}.
Albert Einstein (1915d): Interpreted gravitation as a manifestation of geometry; gave final formulation of the law that governs the dynamic development of the geometry of space with the passage of time.
Albert Einstein (1917): Idealized the universe as a 3 -sphere filled with matter at effectively uniform density; the radius of this 3-sphere could not be envisaged as static without altering his standard 1915 geometrodynamic law; for this reason Einstein introduced a so-called "cosmological term," which he later dropped as "the biggest blunder" in his life [Gamow (1970)].
Formulation by W. de Sitter in 1917 of a cosmological model in which (1) the universe is everywhere isotropic (and therefore homogeneous) and (2) the universe does not change with time, so that the mean density of mass-energy and the mean curvature of space are constant, but in which perforce (3) a cosmological term ("repulsion") of the Einstein type has to be added to balance the attraction of the matter. Observation by de Sitter that he could obtain another static model by removing all the matter from the original model, but that the Lambda\Lambda-term would cause test particles to accelerate away from one another.
From 1917 to 1920, debate about whether spiral nebulae are mere nebulous objects (Harlow Shapley) or are "island universes" or galaxies similar to but external to the Milky Way (H. D. Curtis).
Discovery by Harlow Shapley in 1918, by mapping distribution of about 100 globular clusters of this Galaxy ( 10^(4)10^{4} to 10^(6)10^{6} stars each) in space that center is in direction of Sagittarius (present value of distance from sun ∼10kpc\sim 10 \mathrm{kpc} ).
Independent derivation of evolving homogeneous and isotropic cosmological models [also leading to the relation v=H*v=H \cdot (distance)] by A. Friedmann in 1922 and G. Lemaître in 1927, with Lemaître tieing in his theoretical analysis with the then-ongoing Mt. Wilson work, to become the "father of the big-bang cosmology". (Universe, however, taken to expand smoothly away from Einstein's static Lambda > 0\Lambda>0 solution in Lemaître's original paper).
Remark by H. Weyl in 1923 that test particles in de Sitter model will separate at a rate given by a formula of the form v=H*v=H \cdot (distance).
In 1924, resolution of debate about nature of spiral nebulae by Edwin P. Hubble with Mount Wilson 100-inch telescope; discovery of Cepheid variables in Andromeda and other spiral nebulae, and consequent determination of distances to these nebulae.
Determination by Jan Oort in 1927 of characteristic pattern of radial velocities of stars near sun,
deltav_(r)=Ar cos 2(theta-delta),\delta v_{r}=A r \cos 2(\theta-\delta),
showing that: (1) axis of rotation of stars in Milky Way is perpendicular to disc; (2) sun makes a complete revolution in ∼10^(8)yr\sim 10^{8} \mathrm{yr}; and (3) the effective mass pulling on the sun required to produce a revolution with this period is of the order ∼10^(44)g\sim 10^{44} \mathrm{~g} or ∼10^(11)M_(o.)\sim 10^{11} M_{\odot}.
Age of a uranium ore as established from lead-uranium ratio: greatest value found up to 1927,1.3 xx10^(9)yr1927,1.3 \times 10^{9} \mathrm{yr} (A. Holmes and R. W. Lawson). Age of the lead in the "average" surface rocks of the earth as calculated from time required to produce this lead from the uranium in the same surface rocks, 2xx10^(9)yr2 \times 10^{9} \mathrm{yr} to 6xx10^(9)yr6 \times 10^{9} \mathrm{yr}. Age of elemental uranium as estimated by Rutherford from time required for U^(235)\mathrm{U}^{235} and U^(238)\mathrm{U}^{238} to decay from assumed roughly equal ratio in early days to known very unequal ratio today, ∼3xx10^(9)\sim 3 \times 10^{9} yr.
Establishment by Hubble in 1929 that out to 6xx10^(6)6 \times 10^{6} lyr the velocity of recession of a galaxy is proportional to its distance.
Note by A. S. Eddington in 1930 that Einstein Lambda > 0\Lambda>0 static universe is unstable against any small increase or decrease in the radius of curvature.
Recommendation from Einstein in 1931 hereafter to drop the so-called cosmological term.
Proposal by Einstein and de Sitter in 1932 that one tentatively adopt the simplest assumption that Lambda=0\Lambda=0, that pressure is negligible, and that the reciprocal of the square of the radius of curvature of the universe is neither positive nor negative (spherical or hyperbolic universe) but zero ("cosmologically flat"), thus leading to the relation rho=3H^(2)//8pi\rho=3 H^{2} / 8 \pi (in geometric units).
Evidence uncovered by Grote Reber in 1934 for the existence of a discrete radio source in Cygnus; evidence for this source, Cygnus A, firmed up by J. S. Hey, S. J. Parsons, and J. W. Phillips in 1946; six other discrete radio sources, including Taurus A and Centaurus A, discovered by J. G. Bolton in 1948.
Discovery by E. A. Milne and W. H. McCrea in 1934 of close correspondence between Newtonian dynamics of a large gas cloud and Einstein theory of a dynamic universe, with the scale factor of the expansion satisfying the same equation in both theories, so long as pressure is negligible.
Demonstration by H. P. Robertson and by A. G. Walker, independently, in 1935 that the Lemaitre type of line element provides the most general Riemannian geometry compatible with homogeneity and isotropy.
Classification of nebulae as spiral, barred spiral, elliptical, and irregular by Hubble in 1936.
First detailed theory of thermonuclear energy generation in the sun, H. A. Bethe, 1939.
Box 27.7 (continued)
Reasoning by George Gamow in 1946 that matter in the early universe was dense enough and hot enough to undergo rapid thermonuclear reaction, and that energy densities were radiation-dominated.
Proposal of so-called "steady-state cosmology" by H. Bondi, T. Gold, and F. Hoyle in 1948, lying outside the framework of Einstein's standard general relativity, with "continuous creation of matter" taking place throughout the universe, and the mean age of the matter present being equal to one third of the Hubble time.
Prediction by R. A. Alpher, H. A. Bethe, and G. Gamow in 1948 that the blackbody radiation that originally filled the universe should today have a Planck spectrum corresponding to a temperature of 25 K . Independent conception of same idea by R. H. Dicke in 1964 and start of an experimental search for this primordial cosmic-fireball radiation. Discovery of unwanted and unexpected 7 cm background radiation in 1965 by A. A. Penzias and R. W. Wilson with a temperature of about 3.5 K ; immediate identification of this radiation by Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson as the expected relict radiation.
Radio sources Taurus A, Virgo A, and Centaurus A tentatively and, as it later proved, correctly identified with the Crab Nebula and with the galaxies NGC 4486 and NGC 5128 by J. G. Bolton, G. J. Stanley, and O. B. Slee in 1949.
Analysis by Lemaitre in 1950 of big-bang expansion approaching very closely the Einstein static universe (Lambda > 0)(\Lambda>0) and then, at first slowly, subsequently more and more rapidly, going into exponential expansion.
Discovery by Walter Baade in 1952 that there are two types of Cepheid variables with different period-luminosity relations; consequent increase in Hubble distance scale by factor of about 2.6 , and a corresponding increase in the original value (roughly 2xx10^(9)yr2 \times 10^{9} \mathrm{yr} ) of the Hubble time, H_(o)^(-1)H_{o}^{-1}.
Identification of radio source Cygnus A by W. Baade and R. Minkowski in 1954 with the brightest member of a faint cluster of galaxies, contrary to the then widely held view that the majority of radio sources lie within the Milky Way. Determination of redshift in the optical spectrum of delta lambda//lambda=z=0.057\delta \lambda / \lambda=z=0.057 by Minkowski, implying for Cygnus A a distance of 170 Mpc and a radio luminosity of 10^(45)erg//sec,10^(7)10^{45} \mathrm{erg} / \mathrm{sec}, 10^{7} times the radio power and ten times the optical power of a normal galaxy.
Resolution of radio source Cygnus A in 1956 into two components symmetrically located on either side of the optical galaxy, the first indication that most radio sources are double. Still unsolved is the mystery of the explosion or other mechanism that caused this and other double sources.
Calculation by G. R. Burbidge in 1956 of the kinetic energy in the electrons giving off synchrotron radiation in a radio galaxy and the energy of the magnetic field that holds these electrons in orbit; minimization of the sum of these two energies;
determination that this minimum is of the order of 10^(60)10^{60} ergs (energy of annihilation of half a million suns) for Hercules A, for example.
Solar system determined to have an age of 4.55 xx10^(9)yr4.55 \times 10^{9} \mathrm{yr} or more from relative abundances of Pb^(204,206,207)\mathrm{Pb}^{204,206,207} and U^(235,238)\mathrm{U}^{235,238} in meteorites and oceanic sediments by C. Patterson in 1956; and by others in 1965 and 1969 from evidence on the processes Rb^(87)longrightarrowSr^(87)\mathrm{Rb}^{87} \longrightarrow \mathrm{Sr}^{87} and K^(40)longrightarrowA^(40)\mathrm{K}^{40} \longrightarrow \mathrm{~A}^{40} in meteorites.
Discovery by Allen Sandage in 1958 that what Hubble had identified in distant galaxies as bright stars were H II regions, clumps of hot stars surrounded by a plasma ionized by stars, and consequent upping of Hubble's distance scale by a further factor of about 2.2.
Estimation by Jan Oort in 1958, from luminosity of other galaxies, that matter in galaxies contributes to the density of mass-energy in the universe roughly 3xx10^(-31)g//cm^(3)3 \times 10^{-31} \mathrm{~g} / \mathrm{cm}^{3} [see Peebles (1971) for updated analysis], this being one or two orders of magnitude less than that called for by Einstein's concept that the universe is curved up into closure, and thereby giving rise to "the mystery of the missing matter," the focus of much present-day research.
Discovery of celestial (nonsolar) X-rays in 1962 by Giacconi, Gursky, Paolini, and Rossi. Majority of sources in plane of the Milky Way, presumably local to this galaxy, as is the Crab nebula. Extragalactic sources include the radio galaxy Virgo A and the quasar 3C273.
Revised "3C-catalog" of radio sources published in 1962 by A. S. Bennett, containing 328 sources, nearly complete in coverage between declinations -5^(@)-5^{\circ} and +90^(@)+90^{\circ} for sources brighter than 9 flux units (9xx10^(-26):}\left(9 \times 10^{-26}\right. watt {://m^(2)(Hz))\left./ \mathrm{m}^{2} \mathrm{~Hz}\right) at 178 MHz .
Identification of the first quasistellar object (QSO) by Maarten Schmidt at Mt. Palomar in 1963: radio-position determination of 3C273 to better than 1 second of arc by C. Hazard, M. B. Mackey, and A. J. Shimmins in 1962, followed by Schmidt's taking an optical spectrum of the star-like source and, despite all presumptions that it was a star in this galaxy, trying to fit it, and succeeding, with a redshift of the magnitude (unprecedented for " "star") of delta lambda//lambda=z=0.158\delta \lambda / \lambda=z=0.158. Distance implied by Hubble relation, 1.5 xx10^(9)lyr1.5 \times 10^{9} \mathrm{lyr}; optical brightness, 100 times brightest known galaxy. Largest redshift of any QSO known in 1972, z=2.88z=2.88 (4C05.34; C. R. Lynds). Such a source detectable even if it had a redshift of 3; but no QSO's known in 1972 with such redshifts. See Box 28.1.
Reasoning by Dennis Sciama in 1964 [see also Sciama (1971)] that intergalactic hydrogen can best escape observation if at a temperature between 3xx10^(5)K3 \times 10^{5} \mathrm{~K} and 10^(6)K10^{6} \mathrm{~K}. With as many as 10^(-5)10^{-5} protons and 10^(-5)10^{-5} electrons per cm^(3)\mathrm{cm}^{3} and a temperature lower than 3xx10^(5)K3 \times 10^{5} \mathrm{~K}, the number density of neutral atoms would be great enough and the resulting absorption of Lyman alpha\alpha from a distant galaxy (z=2)(z=2) would be strong enough to show up, contrary to observation.
In 1964 J. E. Gunn and B. A. Peterson, E. J. Wampler, and others determined that, at a temperature greater than 10^(6)K10^{6} \mathrm{~K}, the intensity of 0.25 keV or 50"Å"x50 \AA \mathrm{x}-rays
Box 27.7 (continued)
from intergalactic space would be too high to be compatible with the observations.
Emphasis by Wheeler (1964a) that the dynamic object in Einstein's general relativity is 3-geometry, not 4-geometry, and that this dynamics, both classical and quantum, unrolls in the arena of superspace.
Discovery by Sandage in 1965 of quasistellar galaxies (radio-quiet QSO's).
Discovery by E. M. Burbidge, G. R. Burbidge, C. R. Lynds, and A. N. Stockton in 1965 of a QSO, 3C191, with numerous absorption lines, implying the coexistence of several redshifts in one spectrum.
Fraction (by mass) of matter converted to helium in early few minutes of universe nearly independent of the relative numbers of photons and baryons, over a 10^(6)10^{6} range in values of this number ratio, so long as the universe at 10^(10)K10^{10} \mathrm{~K} is still radiationdominated. Value of this plateau helium abundance (following earlier work of others) first accurately calculated as 27 per cent by P. J. E. Peebles in 1966 and by R. V. Wagoner, W. A. Fowler, and F. Hoyle in 1967.
Proposal by C. W. Misner in 1968 to consider as an important part of early cosmology the anisotropy vibrations of the geometry of space previously brought to attention by E. Kasner and by I. M. Khalatnikov and E. Lifshitz. [Misner's hope to account naturally in this way for the otherwise so puzzling homogeneity of the universe was later dashed.]
Proof on the basis of standard general relativity by S. W. Hawking, G. F. R. Ellis, and R. Penrose in 1968 and 1969 [see also related work of earlier investigators cited in Chapter 44] that a model universe presently expanding and filled with matter and radiation obeying a physically acceptable equation of state must have been singular in the past, however wanting in symmetry it is today.
Discovery of pulsars in 1968 by Hewish, Bell, Pilkington, Scott, and Collins, and their interpretation as spinning neutron stars (see Chapter 24).
"No poet, nor artist of any art, has his complete meaning alone. His significance, his appreciation, is the appreciation of his relation to the dead poets and artists. You cannot value him alone; you must set him, for contrast and comparison, among the dead . . . when a new work of art is created . . . something . . . happens simultaneously to all the works of art which preceded it. The existing monuments form an ideal order among themselves, which is modified by the introduction of the new (the really new) work of art among them."
синетен 28
EVOLUTION OF THE UNIVERSE INTO ITS PRESENT STATE
Cosmology . . . restrains the aberrations of the mere undisciplined imagination.ALFRED NORTH WHITEHEAD (1929, p. 21)
§28.1. THE "STANDARD MODEL" OF THE UNIVERSE
Since the discovery of the cosmic microwave radiation in 1965, extensive theoretical research has produced a fairly detailed picture of how the universe probably evolved into its present state. This picture, called the "standard hot big-bang model" of the universe, is sketched in the present chapter, and its main features appear in Figure 28.1. Gravitation dominates the over-all expansion; but otherwise most details of the evolution are governed much less by gravitation than by the laws of thermodynamics, hydrodynamics, atomic physics, nuclear physics, and high-energy physics. This fact, and the existence of three excellent recent books on the subject [Sciama (1971); Peebles (1972); Zel'dovich and Novikov (1974)], encourage brevity here.
The past evolution of the universe is qualitatively independent of the nature of the homogeneous hypersurfaces (k=-1,0(k=-1,0, or +1 ) and qualitatively independent of the cosmological constant, since the contributions of kk and Lambda\Lambda to the evolution are not important in early stages of the history (small a//a_(o)a / a_{o} ) [see equation (27.40) and Figure 27.5]. One crucial assumption underlies the standard hot big-bang model: that the universe "began" in a state of rapid expansion from a very nearly homogeneous, isotropic condition of infinite (or near infinite) density and temperature.
During the first second after the beginning, according to this analysis, the temperature of the universe was so high that there was complete thermodynamic equilib-
Evolution of universe according to "standard hot big-bang model":
(1) initial state
Figure 28.1.
Evolution of the universe into its present state, according to the standard hot big-bang model. The curves are drawn assuming
but for other values of rho_(mo),rho_(ro)\rho_{m o}, \rho_{r o}, and kk within the limits of observation, the curves are virtually the same (see exercise 28.1). See text and Box 28.1 for detailed discussion of the processes described at the bottom of the figure. [This figure is adapted from Dicke, Peebles, Roll, and Wilkinson (1965).]
(2) thermal equilibrium, decay of particles, recombination of pairs ( 0 < t <= 10sec0<t \leqq 10 \mathrm{sec}.)
(3) decoupling and free propagation of gravitons and neutrinos ( t <= 1sect \leqq 1 \mathrm{sec}.)
rium between photons, neutrinos, electrons, positrons, neutrons, protons, various hyperons and mesons, and perhaps even gravitons (gravitational waves) [see, e.g., Kundt (1971) and references cited therein]. However, by the time the universe was a few seconds old, its temperature had dropped to about 10^(10)K10^{10} \mathrm{~K} and its density was down to ∼10^(5)g//cm^(3)\sim 10^{5} \mathrm{~g} / \mathrm{cm}^{3}; so all nucleon-antinucleon pairs had recombined, all hyperons and mesons had decayed, and all neutrinos and gravitons had decoupled from matter. The universe then consisted of freely propagating neutrinos, and perhaps gravitons, with black-body spectra at temperatures T∼10^(10)KT \sim 10^{10} \mathrm{~K}, plus electron-positron pairs in the process of recombining, plus electrons, neutrons, protons, and photons all in thermal equilibrium at T∼10^(10)KT \sim 10^{10} \mathrm{~K}.
Since that early state, the gravitons (if present) and neutrinos have continued
to propagate freely, maintaining black-body spectra; but their temperatures have been redshifted by the expansion of the universe in accordance with the law
{:(28.1)T prop1//a:}\begin{equation*}
T \propto 1 / a \tag{28.1}
\end{equation*}
(Box 29.2). Consequently, today their temperatures should be roughly 3 K , and they should still fill the universe. Unfortunately, today's technology is far from being able to detect such a "sea" of neutrinos or gravitons. However, if and when they can be detected, they will provide direct observational information about the first one second of the life of the universe!
As the universe continued to expand after the first few seconds, it entered a period lasting from t∼2t \sim 2 seconds to t∼1,000t \sim 1,000 seconds ( T∼10^(10)T \sim 10^{10} to ∼10^(9)K,rho∼10^(+5)\sim 10^{9} \mathrm{~K}, \rho \sim 10^{+5} to 10^(-1)g//cm^(3)10^{-1} \mathrm{~g} / \mathrm{cm}^{3} ), during which primordial element formation occurred. Before this period, there were so many high-energy protons around that they could blast apart any atomic nucleus (e.g., deuterium or tritium or He^(3)\mathrm{He}^{3} or He^(4)\mathrm{He}^{4} ) the moment it formed; after this period, the protons were too cold (had kinetic energies too low) to penetrate each others' coulomb barriers, and all the freely penetrating neutrons from the earlier, hotter stage had decayed into electrons plus protons. Only during the short, crucial period from t∼2t \sim 2 seconds to t∼1,000t \sim 1,000 seconds were conditions right for making elements. Calculations by Gamow (1948), by Alpher and Hermann (1948a, b; 1950), by Fermi and Turkevitch (1950), by Peebles (1966), and by Wagoner, Fowler, and Hoyle (1967) reveal that about 25 per cent of the baryons in the universe should have been converted into He^(4)\mathrm{He}^{4} (alpha particles) during this period, and about 75 per cent should have been left as H^(1)\mathrm{H}^{1} (protons). Traces of deuterium, He^(3)\mathrm{He}^{3}, and Li should have also been created, but essentially no heavy elements. All the heavy elements observed today must have been made later, in stars [see, e.g., Fowler (1967) or Clayton (1968)]. Current astronomical studies of the abundances of the elements give some support for these predictions; but the observational data are not yet very conclusive [see, e.g., Danziger (1968) and pp. 268-275 of Peebles (1971)].
After primordial element formation, the matter and radiation continued to interact thermally through frequent ionization and recombination of atoms, keeping each other at the same temperature. Were the temperatures of radiation and matter not locked together, the radiation would cool more slowly than the matter (for adiabatic expansion, T_(r)prop1//aT_{r} \propto 1 / a, but {:T_(m)prop1//a^(2))\left.T_{m} \propto 1 / a^{2}\right). Thus thermal equilibrium was maintained only by a constant transfer of energy from radiation to matter. But the heat capacity of the radiation was far greater than that of the matter. Therefore the energy transfer had a negligible effect on rho_(r),p_(r)\rho_{r}, p_{r}, and T_(r)T_{r}. It held up the temperature of the matter ( T_(m)=T_(r)T_{m}=T_{r} ) without significantly lowering the temperature of the radiation. On the other hand, the total mass-energy of matter was and is dominated by rest mass. Therefore the energy transfer had negligible influence on rho_(m)\rho_{m}. [This circumstance justifies the approximation of ignoring energy transfer when passing from equation (27.31) to (27.32).]
When the falling temperature reached a few thousand degrees (a//a_(o)∼10^(-3):}\left(a / a_{o} \sim 10^{-3}\right., rho∼10^(-20)g//cm^(3),t∼10^(5)\rho \sim 10^{-20} \mathrm{~g} / \mathrm{cm}^{3}, t \sim 10^{5} years), two things of interest happened: the universe ceased to be radiation-dominated and became matter-dominated [ rho_(m)=rho_(m0)(a_(0)//a)^(3)\rho_{m}=\rho_{m 0}\left(a_{0} / a\right)^{3} came to exceed {:rho_(r)=rho_(ro)(a_(o)//a)^(4)]\left.\rho_{r}=\rho_{r o}\left(a_{o} / a\right)^{4}\right]; and the photons ceased to be energetic enough to keep
(4) primordial element formation ( 2sec.≲t≲1,000sec2 \mathrm{sec} . \lesssim t \lesssim 1,000 \mathrm{sec}.)
(5) thermal interaction of matter and radiation (1,000sec.≲t≲10^(5):}\left(1,000 \mathrm{sec} . \lesssim t \lesssim 10^{5}\right. years)
(6) plasma recombination and transition to matter dominance ( t∼10^(5)yrst \sim 10^{5} \mathrm{yrs}.)
(7) subsequent propagation of photons ( t >= 10^(5)yrst \geq 10^{5} \mathrm{yrs}.)
(8) condensation of stars, galaxies and clusters ( 10^(8)10^{8} yrs. ≲t≲10^(9)\lesssim t \lesssim 10^{9} yrs.)
hydrogen atoms ionized, so the electrons and protons quickly recombined. That these two events were roughly coincident is a result of the specific, nearly conserved value that the entropy per baryon has in our universe:
s-=" entropy per baryon "∼((" number of photons in universe "))/((" number of baryons in universe) ")∼10^(8).s \equiv \text { entropy per baryon } \sim \frac{(\text { number of photons in universe })}{(\text { number of baryons in universe) }} \sim 10^{8} .
Why the universe began with this value of ss, rather than some other value (e.g. unity), nobody has been able to explain.
Recombination of the plasma at t∼10^(5)t \sim 10^{5} years was crucial, because it brought an end to the interaction and thermal equilibrium between radiation and matter ("decoupling"). Thereafter, with very few free electrons off which to scatter, and with Rayleigh scattering off atoms and molecules unimportant, the photons propagated almost freely through space. Unless energy-releasing processes reionized the intergalactic medium sometime between a//a_(o)∼10^(-3)a / a_{o} \sim 10^{-3} and a//a_(o)∼0.1a / a_{o} \sim 0.1, the photons have been propagating freely ever since the plasma recombined. Even if reionization occurred, the photons have been propagating freely at least since a//a_(o)∼0.1a / a_{o} \sim 0.1.
The expansion of the universe has redshifted the temperature of the freely propagating photons in accordance with the equation T prop1//aT \propto 1 / a (see Box 29.2). As a consequence, today they have a black-body spectrum with a temperature of 2.7 K . They are identified with the cosmic microwave radiation that was discovered in 1965, and they give one direct information about the nature of the universe at the time they last interacted with matter ( a//a_(o)∼10^(-3),t∼10^(5)a / a_{o} \sim 10^{-3}, t \sim 10^{5} years if reionization did not occur; a//a_(o)∼0.1,t∼5xx10^(8)a / a_{o} \sim 0.1, t \sim 5 \times 10^{8} years if reionization did occur.)
Return to the history of matter. Before plasma recombination, the photon pressure ("elasticity of the cosmological fluid") prevented the uniform matter ( 25 per cent He^(4),75\mathrm{He}^{4}, 75 per cent H) from condensing into stars, galaxies, or clusters of galaxies. However, after recombination, the photon pressure was gone, and condensation could begin. Small perturbations in the matter density, perhaps dating back to the beginning of expansion, then began to grow larger and larger. Somewhere between a//a_(o)∼1//30a / a_{o} \sim 1 / 30 and a//a_(o)∼1//10(10^(8):}a / a_{o} \sim 1 / 10\left(10^{8}\right. years ≲t≲10^(9)\lesssim t \lesssim 10^{9} years )) these perturbations began developing into stars, galaxies, and clusters of galaxies. Slightly later, at a//a_(o)∼1//4a / a_{o} \sim 1 / 4, quasars probably "turned on," emitting light which astronomers now receive at Earth (see Box 28.1).
EXERCISE
Exercise 28.1. UNCERTAINTY IN EVOLUTION
Current observations, plus the assumption of complete homogeneity and isotropy at the beginning of expansion, plus the assumption that the excess of leptons over antileptons is less than or of the order of the excess of baryons over antibaryons, place the following limits on the cosmological parameters today:
Matter density today =rho_(m0)=\rho_{m 0}, between 10^(-28)10^{-28} and 2xx10^(-31)g//cm^(3)2 \times 10^{-31} \mathrm{~g} / \mathrm{cm}^{3}; k=0k=0 or +1 or -1 ;
temperature of electromagnetic radiation today =2.7+-0.1K=2.7 \pm 0.1 \mathrm{~K}.
Total radiation density [observed photons, plus neutrinos and gravitons that presumably originated in big bang in thermal equilibrium with photons ]=rho_("ro ")]=\rho_{\text {ro }}, between 0.7 xx10^(-33)0.7 \times 10^{-33} and 1.2 xx10^(-33)g//cm^(3)1.2 \times 10^{-33} \mathrm{~g} / \mathrm{cm}^{3}.
(continued on page 769)
Box 28.1 EVOLUTION OF THE QUASAR POPULATION
That the large-scale, average properties of the universe are changing markedly with time one can infer from quasar data. In brief, there appear to have been about 50 times more quasars in the universe at a redshift z~~2z \approx 2 than at z~~0.5z \approx 0.5; and there may well have been fewer, or none, at redshifts z > 3z>3. (On the use of redshift to characterize time since the big bang, see Box 29.3.) In greater detail, Schmidt (1972) gives the following analysis of the data:*
Schmidt assumes from the outset that quasar redshifts are cosmological in origin [redshift =(=( Hubble constant )xx) \times (distance );$29.2]) ; \$ 29.2]. The evidence for this is
a. Observational: Some quasars are located in clusters of galaxies [as evidenced both by position on sky and by quasar having same redshift as galaxies in cluster; see Gunn (1971)]. Since the evidence for the cosmological distanceredshift relation for galaxies is overwhelming (Boxes 29.4 and 29.5), the redshifts of these particular quasars must be cosmological.
b. Theoretical: Observed quasar redshifts of z∼1z \sim 1 to 3 cannot be gravitational in origin; objects with gravitational redshifts larger than z~~0.5z \approx 0.5 are unstable against collapse (see Chapters 24 and 26 and Box 25.9). Nor are the quasar redshifts likely to be Doppler; how could so massive an object be accelerated to v~~1v \approx 1 without complete disruption? The only remaining possibility is a cosmological redshift. For this reason, opponents of the cosmological hypothesis usually feel pressed to invoke in the quasars a breakdown of the laws of physics as one understands them today. [See, e.g., Arp (1971) and references cited therein. These references also describe evidence against the cosmological assumption, evidence that a few prominent investigators find compelling, but that most do not as of 1972.]
Schmidt then asks how many quasars, NN, there were in the universe at a time corresponding to the redshift zz, and with absolute luminosity per unit frequency, L_(v)(2,500"Å")L_{v}(2,500 \AA) at the wavelength 2500"Å"2500 \AA as measured in the quasar's local Lorentz frame.
The data on quasars available in 1972 are not at all sufficient to determine N[z,L_(nu)(2,500"Å")]N\left[z, L_{\nu}(2,500 \AA)\right] uniquely. But they are sufficient to show unequivocally that: a. There must have been evolution; N(z,L_(p))N\left(z, L_{p}\right) cannot be independent of zz.
b. The evolution cannot have resided primarily in the luminosities: the total number of quasars,
must have changed markedly with time (with zz ).
*Our version of Schmidt's (1972) argument is oversimplified. The reader interested in greater precision should consult his original paper.
Box 28.1 (continued)
c. If the evolution was primarily in the total number, N_("tot ")(z)N_{\text {tot }}(z), i.e., if the changes in the relative luminosity distribution at 2,500"Å"2,500 \AA
This steep increase in number as one goes backward in time-and all other basic features of the observed quasar redshift and magnitude distributions for z≲2z \lesssim 2-can be fit in a universe with sigma_(0)=q_(0)=1\sigma_{0}=q_{0}=1 by either of the evolution laws
Here t_(0)t_{0} is the current age of the universe and tt was the age at redshift zz.
d. These evolution laws, when extrapolated beyond a redshift z~~2z \approx 2 and when combined with the observed relative luminosity function f(z,L_(p))f\left(z, L_{p}\right) for quasars near apparent magnitude 18, predict that an observer on Earth should see the following fractions of nineteenth and twentieth-magnitude quasars to have redshifts greater than 2.5 :
In 1972 about 30 quasars fainter than m=18.5m=18.5 are known, and of these only 1(3%)1(3 \%) has z > 2.5z>2.5. This shows, in Schmidt's words, "that the density law (1+z)^(6)(1+z)^{6} cannot persist beyond a redshift of around 2.5." Schmidt regards the 10^(5(t_(0)-t)//t_(0))10^{5\left(t_{0}-t\right) / t_{0}} law (which becomes nearly constant at z > 2.5z>2.5 ) to be also in apparent conflict with the observations, but he says that "further spectroscopic work on faint quasars is needed to confirm this suspicion."
One reason for caution is the difficult problem of removing "observational selection effects" from the data. Schmidt, Sandage, and others have independently searched for selection effects that might produce an artificial apparent decrease in the number of quasars at z > 2.5z>2.5. None have been found. In the words of Sandage (1972d) "The apparent cutoff in quasar redshifts near z=2.8z=2.8 [has been] examined for selection effects that could produce it artificially. If the cutoff is real, it may be the time of the birth of the first quasars, although the suggested redshift is unexpectedly small. At z=3z=3 in a q_(0)=1q_{0}=1 universe, the look-back time is 89 per cent of the Friedmann age. Assessment of the observational selection effects shows that none are positively established that could produce the cutoff artificially."
(The uncertainties taken into account in rho_(ro)\rho_{r o} are uncertainty about whether quadrupole moments at early times were sufficient to create gravitons at the full level corresponding to thermal equilibrium, and uncertainty about the number and statistical weights of particle species in equilibrium at the time gravitons decoupled.) Use the equations in $27.10\$ 27.10 to calculate the uncertainties in the evolutionary history (Figure 28.1) caused by these uncertainties in the present state of the universe.
§28.2. STANDARD MODEL MODIFIED FOR PRIMORDIAL CHAOS
The standard hot big-bang model is remarkably powerful and accords well with observations (primordial helium abundances; existence, temperature, and isotropy of cosmic microwave radiation; homogeneity and isotropy of universe in the large; close accord between age of universe as measured by expansion and ages of oldest stars; . . .). However, in 1972 it encounters apparent difficulty with one item: the origin of galaxies. In a universe that is initially homogeneous and isotropic it is not clear that random fluctuations will give rise (after plasma recombination) to perturbations in the density of matter of sufficient amplitude to condense into galaxies. The perturbations that eventually form galaxies might have to reside in the initial, exploding state of the universe. [See Zel'dovich and Novikov (1974) for detailed review and discussion; see also references cited in §30.1.]
Is it reasonable to assume a small amount of initial inhomogeneity? Is it not much more reasonable to assume either perfect homogeneity (one extreme) or perfect chaos (the other extreme)?
Thus, if perfect initial homogeneity turns out to be incompatible with the origin of galaxies, it is attractive to try "perfect initial chaos"-i.e., completely random initial conditions, with a full spectrum of fluctuations in density, entropy, and local expansion rate [Misner (1968,1969 b)](1968,1969 b)]. It is conceivable, but far from proved, that during its subsequent evolution such a model universe will quickly smooth itself out by natural processes (Chapter 30) such as "Mixmaster oscillations," neutrinoinduced viscosity [see, e.g., Matzner and Misner (1971)], and gravitational curva-ture-induced creation of particle pairs [Zel'dovich (1972)]. Will one be left, after a few seconds or less, with a nearly homogeneous and isotropic, Friedmann universe, containing just enough remaining perturbations to condense eventually into galaxies? Theoretical calculations have not yet been carried far enough to give a clear answer. Of course, after the initial chaos subsides, if it subsides, such a model universe will evolve in accord with the standard big-bang model of the last section.
§28.3. WHAT "PRECEDED" THE INITIAL SINGULARITY?
No problem of cosmology digs more deeply into the foundations of physics than the question of what "preceded" the "initial state" of infinite (or near infinite) density, pressure, and temperature. And, unfortunately, no problem is farther from solution in 1973.
What if the universe began chaotic?
The initial singularity and quantum gravitational effects
General relativity predicts, inexorably, that even if the "initial state" was chaotic rather than smooth, it must have involved a spacetime "singularity" of some sort [see Hawking and Ellis (1968); also §34.6\S 34.6 of this book]. And general relativity is incapable of projecting backward through the singularity to say what "preceded" it. Perhaps only by coming to grip with quantum gravitational effects (marriage of quantum theory with classical geometrodynamics) will one ever reach a clear understanding of the initial state and of what, if anything, "preceded" it [see Misner (1969c), Wheeler (1971c)]. For further discussion of these deep issues, see $§34.6\$ \S 34.6, 43.4, the final section of Box 30.1, and Chapter 44.
§28.4. OTHER COSMOLOGICAL THEORIES
This book confines attention to the cosmology of general relativity. If one were to abandon general relativity, one would have a much wider set of possibilities, including (1) the steady-state theory [Hoyle (1948); Bondi and Gold (1948)], which has not succeeded in accounting for the cosmic microwave radiation or in explaining observed evolutionary effects in radio sources and quasars [Box 28.1]; (2) the KleinAlfvén "hierarchic cosmology" of matter in an asymptotically flat spacetime [Alfvén and Klein (1962), Alfvén (1971), Klein (1971), Moritz (1969), de Vaucouleurs (1971)], which disagrees with cosmic-ray and gamma-ray observations [Steigman (1971)]; and the Brans-Dicke cosmologies [Dicke (1968), Greenstein (1968a,b), Morganstern (1973)], which are qualitatively the same and quantitatively almost the same as the standard hot big-bang model. However, no motivation or justification is evident for abandoning general relativity. The experimental basis of general relativity has been strengthened substantially in the past decade (Chapters 38-40); and the standard big-bang model of the universe predicted by general relativity accords remarkably well with observations-far better than any other model ever proposed!
снартев 29
PRESENT STATE AND FUTURE EVOLUTION OF THE UNIVERSE
§29.1. PARAMETERS THAT DETERMINE THE FATE OF THE UNIVERSE
Will the universe continue to expand forever; or will it slow to a halt, reverse into contraction, and implode back to a state of infinite (or near infinite) density, pressure, temperature, and curvature? The answer is not yet known for certain. To discover the answer is one of the central problems of cosmology today.
The only known way to discover the answer is to measure, observationally, the present state of the universe; and then to calculate the future evolution using Einstein's field equations. The field equations have already been solved in $§27.10\$ \S 27.10 and 27.11. From those solutions one reads off the following correlation between the present state of the universe and its future.
If Lambda=0\Lambda=0 [in accord with Einstein's firmly held principle of simplicity]:
Expansion forever Longleftrightarrow\Longleftrightarrow negative or zero spatial curvature for hypersurfaces of homogeneity, i.e., k//a_(o)^(2) <= 0k / a_{o}{ }^{2} \leq 0 ("open" or "flat");
Recontraction Longleftrightarrow\Longleftrightarrow positive spatial curvature for homogeneous hypersurfaces, i.e., k//a_(o)^(2) > 0k / a_{o}^{2}>0 ("closed").
If Lambda!=0\Lambda \neq 0 :
Expansion forever Longleftrightarrow Lambda >= Lambda_("crit ")-={[0," if "k <= 0","],[(4pirho_(mo)a_(o))^(-2)," if "k > 0;]:}\Longleftrightarrow \Lambda \geq \Lambda_{\text {crit }} \equiv \begin{cases}0 & \text { if } k \leq 0, \\ \left(4 \pi \rho_{m o} a_{o}\right)^{-2} & \text { if } k>0 ;\end{cases}
Recontraction Longleftrightarrow Lambda < Lambda_("crit ")\Longleftrightarrow \Lambda<\Lambda_{\text {crit }}.
Evidently three parameters are required to predict the future: the cosmological constant, Lambda\Lambda; the curvature parameter today for the hypersurface of homogeneity, k//a_(o)^(2)k / a_{o}{ }^{2}; and the density of matter today, rho_(mo)\rho_{m o}. (To extrapolate into the past, as was done in the last chapter, one needs, besides these quantities, the radiation density today, rho_(ro)\rho_{r o}. But rho_(ro)\rho_{r o} is so small now and is getting smaller so fast ( rho_(r)propa^(-4);rho_(m)propa^(-3)\rho_{r} \propto a^{-4} ; \rho_{m} \propto a^{-3} ) that it can have no influence on the decision between the possibilities just listed.
This chapter is entirely Track 2. Chapter 27 (idealized cosmological models) is needed as preparation for it, but this chapter is not needed as preparation for any later chapter.
Expansion forever vs. recontraction of universe
Parameters required to predict future of universe:
(1) "relativity parameters" Lambda,k//a_(o)^(2),rho_(mo)\Lambda, k / a_{o}{ }^{2}, \rho_{m o}
(2) "observational parameters" H_(o),q_(o),sigma_(o)H_{o}, q_{o}, \sigma_{o}
(3) relationship between relativity parameters and observational parameters
The task of predicting the future, then, reduces to the task of measuring the "relativity parameters" Lambda,k//a_(0)^(2)\Lambda, k / a_{0}{ }^{2}, and rho_(mo)\rho_{m o}.
In tackling this task, observational cosmologists prefer to replace the three "relativity parameters," which have immediate significance for relativity theory, by parameters that are more directly observable. One parameter close to the observations is the Hubble expansion rate today, i.e., the "Hubble constant,"
The relationships between these three "observational parameters" and the three "relativity parameters" Lambda,k//a_(o)^(2)\Lambda, k / a_{o}{ }^{2}, and rho_(mo)\rho_{m o} (together making six "cosmological parameters") can be calculated by combining definitions (29.1) with the Einstein field equations (27.39), which, evaluated today, say
By combining these equations, one finds the relationships shown in Box 29.1, where the implications of several values of sigma_(o)\sigma_{o} and q_(o)q_{o} are also shown.
EXERCISE
Exercise 29.1. IMPLICATIONS OF PARAMETER VALUES
Derive the results quoted in Box 29.1.
§29.2. COSMOLOGICAL REDSHIFT
One of the key pieces of observational data used in measurements of H_(o),q_(o)H_{o}, q_{o}, and sigma_(o)\sigma_{o} is the cosmological redshift: spectral lines emitted by galaxies far from Earth and received at Earth are found to be shifted in wavelength toward the red. For example, the [O II] 3727 line, when both emitted and observed in an Earth-bound laboratory, has a wavelength of 3727"Å"3727 \AA. However, when it is emitted by a star in the galaxy
Box 29.1 OBSERVATIONAL PARAMETERS COMPARED TO RELATIVITY PARAMETERS
A. Relativity Parameters
Matter density today,
rho_(mo)\rho_{m o}
Curvature of hypersurface of homogeneity today,
k//a_(o)^(2)k / a_{o}{ }^{2}
Cosmological constant,
Lambda\Lambda
Radiation density today, rho_(ro)\rho_{r o} (unimportant for the present dynamics of the universe, and therefore ignored in this chapter)
If Lambda!=0\Lambda \neq 0
(a) sigma_(o) > (1)/(3)(q_(o)+1)Longleftrightarrow k > 0quad((" positive curvature; ")/(" "closed" universe "))\sigma_{o}>\frac{1}{3}\left(q_{o}+1\right) \Longleftrightarrow k>0 \quad\binom{\text { positive curvature; }}{\text { "closed" universe }},
and in this case, sigma_(o)-q_(o) >= (1)/(sigma_(o)^(2))(sigma_(o)-(q_(o)+1)/(3))^(3)Longleftrightarrow\sigma_{o}-q_{o} \geq \frac{1}{\sigma_{o}^{2}}\left(\sigma_{o}-\frac{q_{o}+1}{3}\right)^{3} \Longleftrightarrow universe will expand forever, sigma_(o)-q_(o) < (1)/(sigma_(o)^(2))(sigma_(o)-(q_(o)+1)/(3))^(3)Longleftrightarrow\sigma_{o}-q_{o}<\frac{1}{\sigma_{o}{ }^{2}}\left(\sigma_{o}-\frac{q_{o}+1}{3}\right)^{3} \Longleftrightarrow universe will eventually recontract;
(b) sigma_(o)=(1)/(3)(q_(o)+1)Longleftrightarrow k=0\sigma_{o}=\frac{1}{3}\left(q_{o}+1\right) \Longleftrightarrow k=0((" zero curvature; ")/(" "flat" universe "))\binom{\text { zero curvature; }}{\text { "flat" universe }},
and in this case, sigma_(o) >= q_(o)Longleftrightarrow\sigma_{o} \geq q_{o} \Longleftrightarrow universe will expand forever, sigma_(o) < q_(o)Longleftrightarrow\sigma_{o}<q_{o} \Longleftrightarrow universe will eventually recontract;
(c) sigma_(o) < (1)/(3)(q_(o)+1)Longleftrightarrow k < 0\sigma_{o}<\frac{1}{3}\left(q_{o}+1\right) \Longleftrightarrow k<0 ((" negative curvature; ")/(" "open" universe "))\binom{\text { negative curvature; }}{\text { "open" universe }},
and in this case, sigma_(o) >= q_(o)Longleftrightarrow\sigma_{o} \geq q_{o} \Longleftrightarrow universe will expand forever, sigma_(o) < q_(o)<=>\sigma_{o}<q_{o} \Leftrightarrow universe will eventually recontract.
3C 295 (presumably with the same wavelength, lambda_("em ")=3727"Å"\lambda_{\text {em }}=3727 \AA ) and received at Earth, it is measured here to have the wavelength lambda_("rec ")=5447"Å"\lambda_{\text {rec }}=5447 \AA. The fractional change in wavelength is
{:(29.3)z-=(lambda_(rec)-lambda_(em))//lambda_(em)=0.4614" for "3C295:}\begin{equation*}
z \equiv\left(\lambda_{\mathrm{rec}}-\lambda_{\mathrm{em}}\right) / \lambda_{\mathrm{em}}=0.4614 \text { for } 3 \mathrm{C} 295 \tag{29.3}
\end{equation*}
The cosmological redshift is observed to affect all spectral lines alike, and not only lines in the visible spectrum. Thus, the 21-cm21-\mathrm{cm} line of hydrogen, with 400,000 times the wavelength of the central region of the visible, undergoes a redshift that agrees (within the errors of the measurements) with the redshifts of lines in the visible for recession velocities of the order of v∼0.005v \sim 0.005, according to observation of thirty objects by Dieter, Epstein, Lilley, and Roberts (1962) and further observations by Roberts (1965).
No one has ever put forward a satisfactory explanation for the cosmological redshift other than the expansion of the universe (see below). The idea has been proposed at various times by various authors that some new process is at work ("tired light") in which photons interact with atoms or electrons on their way from source to receptor, and thereby lose bits and pieces of their energy. Ya. B. Zel'dovich (1963) gives a penetrating analysis of the difficulties with any such ideas:
(1) "If the energy loss is caused by an interaction with the intergalactic matter, it is accompanied by a transfer of momentum; that is, there is a change of the direction of motion of the photon. There would then be a smearing out of images; a distant star would be seen as a disc, not a point, and that is not what is observed." (2) "Let us suppose that the photon decays, gamma longrightarrowgamma^(')+k\gamma \longrightarrow \gamma^{\prime}+k, giving up a small part of its energy to some particle, kk. It follows from the conservation laws that kk must move in the direction of the photon (this, by the way, avoids a smearing out), and must have zero rest mass. Because of the statistical nature of the process, however, some photons would lose more energy than others, and there would be a spectral broadening of the lines, which is also not observed."
(3) If there does exist any such decay process, then simple arguments of special relativity that Zel'dovich attributes to M. P. Bronshtein, and spells out in detail, demand the relationship
([" probability per "],[" second of "],[" "photon decay" "])=(((" a universal constant with ")/(" the dimensions "sec^(-2))))/((" frequency of photon in "sec^(-1)))\left(\begin{array}{l}
\text { probability per } \\
\text { second of } \\
\text { "photon decay" }
\end{array}\right)=\frac{\binom{\text { a universal constant with }}{\text { the dimensions } \sec ^{-2}}}{\left(\text { frequency of photon in } \sec ^{-1}\right)}
"Thus," Zel'dovich concludes, "if the decay of photons is possible at all, those in radio waves must decay especially rapidly! This would mean that the Maxwell equation for a static electric field would have to be changed . . . . There is no experimental indication of such effects: the radio-frequency radiation from distant sources is transmitted to us not a bit more poorly than visible light, and the red shift measured in different parts of the spectrum is exactly the same . . . Thus, suggestions that there is an explanation of the red shift other than Friedmann's fail completely."
Figure 29.1.
Redshift as an effect of standing waves. The ratio of wavelengths, lambda_(rec)//lambda_(em)\lambda_{\mathrm{rec}} / \lambda_{\mathrm{em}}, is identical with the ratio of dimensions, a_(rec)//a_(em)a_{\mathrm{rec}} / a_{\mathrm{em}} in any closed spherically symmetrical (Friedmann) model universe. The atom excites an nn-node standing wave in the universe. The number nn stays constant during the expansion. Therefore wavelengths increase in the same proportion as the dimensions of the universe. One sees immediately in this way that the redshift is independent of all such details as (1) why the expansion came about (spherical symmetry, but arbitrary equation of state); (2) the rate-uniform or nonuni-form-at which it came about; and (3) the distance between source and receptor at emission, at reception, or at any time in-between. The reasoning in the diagram appears to depend on the closure of the universe (standing waves; k=+1k=+1 rather than 0 or -1 ). That closure is not required for this simple result is seen from the further analysis given in the text.
Not the least among the considerations that lead one to accept the general recession of the galaxies as the explanation for the redshift is the circumstance that this general recession was predicted [Friedmann (1922)] before the redshift was observed [Hubble (1929)].
The cosmological redshift is easily understood (Figure 29.1) in terms of the standard big-bang model for the universe. A detailed analysis focuses attention on three processes: emission of the light, propagation of the light through curved spacetime from emitter to receiver, and reception of the light. Emission and reception occur in the proper reference frames (orthonormal tetrads) of the emitter and receiver; they are special-relativistic phenomena. Propagation, by contrast, is a general-relativistic process; it is governed by the law of geodesic motion in curved spacetime.
In calculating all three processes-emission, propagation, and absorption-one
needs a coordinate system. Use the coordinates (t,chi,theta,phi)(t, \chi, \theta, \phi) or ( eta,chi,theta,phi\eta, \chi, \theta, \phi ) introduced in Chapter 27; and orient the space coordinates in such a way that the paths of the light rays through the coordinate system are simple. This is best done by putting the origin of the coordinate system (chi=0)(\chi=0) at the Earth. Then the emitting galaxy will lie at some "radius" chi_(e)\chi_{e} and some angular position (theta_(e),phi_(e))\left(\theta_{e}, \phi_{e}\right). The cosmological line element
{:[ds^(2)=-dt^(2)+a^(2)(t)[dchi^(2)+Sigma^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]],[(29.4a)=a^(2)(eta)[-deta^(2)+dchi^(2)+Sigma^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]","],[(29.4b)Sigma={[sin chi," if "k=+1","],[chi," if "k=0","],[sinh chi," if "k=-1","]:}]:}\begin{align*}
d s^{2} & =-d t^{2}+a^{2}(t)\left[d \chi^{2}+\Sigma^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \\
& =a^{2}(\eta)\left[-d \eta^{2}+d \chi^{2}+\Sigma^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right], \tag{29.4a}\\
& \Sigma= \begin{cases}\sin \chi & \text { if } k=+1, \\
\chi & \text { if } k=0, \\
\sinh \chi & \text { if } k=-1,\end{cases} \tag{29.4b}
\end{align*}
is spherically symmetric about chi=0\chi=0 (i.e., about the Earth) whether k=-1,0k=-1,0, or +1 . Consequently, the geodesics (photon world lines) that pass through both Earth and the emitting galaxy must all be radial
(One who wishes to forego any appeal to symmetry can examine the geodesic equation in the (t,chi,theta,phi)(t, \chi, \theta, \phi) coordinate system, and discover that if d theta//d lambda=d \theta / d \lambda=d phi//d lambda=0d \phi / d \lambda=0, then d^(2)theta//dlambda^(2)=d^(2)phi//dlambda^(2)=0d^{2} \theta / d \lambda^{2}=d^{2} \phi / d \lambda^{2}=0. Consequently a geodesic that is initially radial will always remain radial.)
Consider, now, emission. A galaxy at rest (moving with the "cosmological fluid") at ( chi_(e),theta_(e),phi_(e)\chi_{e}, \theta_{e}, \phi_{e} ) emits two successive crests, AA and BB, of a wave train toward Earth at coordinate times t_(Ae)t_{A e} and t_(Be)t_{B e}. It has been arranged that proper time as measured on the galaxy is the same as coordinate time ( t=tau+t=\tau+ const. was part of the construction process for the coordinate system in §27.4). Consequently the period of the radiation as seen by the emitter is P_(em)=t_(eB)-t_(eA)P_{e m}=t_{e B}-t_{e A}; and the wavelength is the same as the period when geometrized units are used:
Next examine propagation. Wave crests AA and BB propagate along null geodesics. This fact enables one to read the world lines of the wave crests, chi_(A)(t)\chi_{A}(t) and chi_(B)(t)\chi_{B}(t), directly from the line element (29.4): ds^(2)=0d s^{2}=0 guarantees that a(t)d chi=-dt(-a(t) d \chi=-d t(-, not + , because the light propagates toward the Earth at chi=0\chi=0 ). Consequently, the world lines are
{:[(29.7)chi_(e)-chi_(A)(t" or "eta)=eta-eta_(eA)=int_(t_(eA))^(t)a^(-1)dt","],[chi_(e)-chi_(B)(t" or "eta)=eta-eta_(eB)=int_(t_(eB))^(t)a^(-1)dt.]:}\begin{align*}
& \chi_{e}-\chi_{A}(t \text { or } \eta)=\eta-\eta_{e A}=\int_{t_{e A}}^{t} a^{-1} d t, \tag{29.7}\\
& \chi_{e}-\chi_{B}(t \text { or } \eta)=\eta-\eta_{e B}=\int_{t_{e B}}^{t} a^{-1} d t .
\end{align*}
Finally, examine reception. The receiver on Earth moves with the "cosmological fluid," just as does the distant emitter. (Ignore the Earth's "peculiar motion" relative
to the fluid-motion around the sun, motion around center of our Galaxy, etc.; it can be taken into account by an ordinary Doppler correction.) Thus, for receiver as for emitter, proper time is the same as coordinate time, and
where t_(rB)t_{r B} and t_(rA)t_{r A} are the times of reception of the successive wave crests.
Now combine equations (29.6), (29.7), and (29.8) to obtain the redshift. The receiver is at chi=0\chi=0. Therefore equations (29.7) say
{:[0=chi_(e)-int_(t_(eA))^(t_(rA))a^(-1)dt],[(29.9)0=chi_(e)-int_(t_(eB))^(t_(rB))a^(-1)dt]:}\begin{align*}
& 0=\chi_{e}-\int_{t_{e A}}^{t_{r A}} a^{-1} d t \\
& 0=\chi_{e}-\int_{t_{e B}}^{t_{r B}} a^{-1} d t \tag{29.9}
\end{align*}
Subtract these equations from each other to obtain
{:[0=int_(t_(eB))^(t_(rB))a^(-1)dt-int_(t_(eA))^(t_(rA))a^(-1)dt],[=int_(t_(rA))^(t_(rB))a^(-1)dt-int_(t_(eA))^(t_(eB))a^(-1)dt~~(t_(rB)-t_(rA))/(a(t_(r)))-(t_(eB)-t_(eA))/(a(t_(e)));]:}\begin{aligned}
0 & =\int_{t_{e B}}^{t_{r B}} a^{-1} d t-\int_{t_{e A}}^{t_{r A}} a^{-1} d t \\
& =\int_{t_{r A}}^{t_{r B}} a^{-1} d t-\int_{t_{e A}}^{t_{e B}} a^{-1} d t \approx \frac{t_{r B}-t_{r A}}{a\left(t_{r}\right)}-\frac{t_{e B}-t_{e A}}{a\left(t_{e}\right)} ;
\end{aligned}
These redshift equations confirm the simple result of Figure 29.1: As the light ray propagates, its wavelength (as measured by observers moving with the "fluid") increases in direct proportion to the linear expansion of the universe. The ratio of the wavelength to the expansion factor, lambda//a\lambda / a, remains constant. For important applications of this result, see Boxes 29.2 and 29.3.
EXERCISES
Exercise 29.2. ALTERNATIVE DERIVATION OF REDSHIFT
Notice that the only part of the line element that is relevant for the light ray is
ds^(2)=-dt^(2)+a^(2)(t)dchi^(2)d s^{2}=-d t^{2}+a^{2}(t) d \chi^{2}
since d theta=d phi=0d \theta=d \phi=0 along its world line (spherical symmetry!). Regard the light ray as made
Box 29.2 COSMOLOGICAL REDSHIFT OF THE PRIMORDIAL RADIATION
As an important application of the redshift formula
[equation (29.10)], consider the radiation emerging from the hot big bang. Because it is initially in thermal equilibrium with matter, this primordial radiation initially has a Planck black-body spectrum. Subsequent interactions with matter cannot change the spectrum, because the matter remains in thermal equilibrium with the radiation so long as interactions are occurring. The cosmological redshift can and does change the spectrum, however. It was shown in exercise 22.17, using kinetic theory, that radiation with a Planck spectrum as viewed by one observer has a Planck spectrum as viewed by all observers; but the observed temperature is redshifted in precisely the same manner as the frequency of an individual photon is redshifted. Consequently, as seen by observers at rest
in the "fluid," the temperature of the primordial radiation is redshifted
{:(2)T prop1//a.:}\begin{equation*}
T \propto 1 / a . \tag{2}
\end{equation*}
This is true after plasma recombination, when the radiation and matter are decoupled, as well as before recombination, when they are interacting. And it is true not only for the primordial photons but also for thermalized neutrinos and gravitons emerging from the hot big bang.
There is another way to derive the redshift equation (2). Combine the equation
for the decrease of energy density with adiabatic expansion.
Box 29.3 USE OF REDSHIFT TO CHARACTERIZE DISTANCES AND TIME
Distance: When discussing objects within the Earth's cluster of galaxies, astronomers typically describe distances in units of lightyears or parsecs. But when dealing with more distant objects (galaxies, quasars, etc.), astronomers find it more convenient to describe distance in terms of what is actually observed: redshift. For example, the statement "the galaxy 3C 295 is at a redshift of 0.4614 " means that "3C 295 is at that distance from Earth [given by equation (29.16)] which corresponds to a redshift of z=0.4614z=0.4614."
Time: When discussing events that occurred during the last few 10^(9)10^{9} years, astronomers usually measure time in units of years. Example: "The solar system condensed out of interstellar gas 4.6 xx10^(9)4.6 \times 10^{9} years ago" [see Wasserburg and Burnett (1968)]. But when dealing with events much nearer the beginning of the universe, all of which have
essentially the same age, of about 12 xx10^(9)12 \times 10^{9} years, astronomers find it more convenient to describe time in terms of redshift. Example: "The primordial plasma recombined at a redshift of 1,000 " means that "If a photon had been emitted at the time of plasma recombination, and had propagated freely ever since, it would have experienced a total redshift between then and now of z=1,000z=1,000." Equivalently, since 1+z=(a_(o)//a)1+z=\left(a_{o} / a\right) [see equation (29.11)], "the plasma recombined when the universe was a factor of 1+z~~1,0001+z \approx 1,000 smaller than it is today." [Application: In Figure 28.1, where the past evolution of the universe is summarized, one can freely replace the horizontal scale a//a_(o)a / a_{o} by 1//(1+z)1 /(1+z), and thereby see that primordial element formation occurred at a redshift of z~~10^(9)z \approx 10^{9}.) The conversion from redshift units to time units is strongly dependent on the parameters rho_(mo),rho_(ro)\rho_{m o}, \rho_{r o}, and k//a_(o)^(2)k / a_{o}{ }^{2} [see $$27.10\$ \$ 27.10 and 27.11 ; also equation (29.15)].
of photons with 4-momenta p\boldsymbol{p}. From the geodesic equation (or, for the reader who has studied chapter 25 , from arguments about Killing vectors), show that
is conserved along the photon's world line. Use this fact, the fact that a photon's 4 -momentum is null, p*p=0\boldsymbol{p} \cdot \boldsymbol{p}=0, and the equation E=-p*uE=-\boldsymbol{p} \cdot \boldsymbol{u} for the energy measured by an observer with 4 -velocity u\boldsymbol{u}, to derive the redshift equation (29.11).
Exercise 29.3. REDSHIFT OF PARTICLE DE BROGLIE WAVELENGTHS
A particle of finite rest mass mu\mu moves along a geodesic world line through the expanding cosmological fluid. Let
be the spatial 4-momentum of the particle as measured by observers at rest in the fluid. (The ordinary velocity they measure in their proper reference frames is vv.) The associated "de Broglie wavelength" of the particle is lambda-=h//p\lambda \equiv h / p.
(a) Show that this de Broglie wavelength is redshifted in precisely the same manner as a photon wavelength:
(b) Employing this result, show that, for the molecules of an ideal gas that fills the universe, their mean kinetic energy decreases in inverse proportion to a^(2)a^{2} when the gas is nonrelativistic and (like photon energies) in inverse proportion to aa when the gas is highly relativistic.
Derivation of distance-redshift relation
§29.3. THE DISTANCE-REDSHIFT RELATION; MEASUREMENT OF THE HUBBLE CONSTANT
Equation (29.11) expresses the redshift in terms of the change in expansion factor between the event of emission and the event of reception. For "nearby" emitters (emitters at distances much less than 1//H_(o)1 / H_{o}, the "Hubble length") it is more convenient to express the redshift in terms of the distance between the emitter and Earth. That distance ("present distance") is defined on the hypersurface of homogeneity that passes through Earth today, since that hypersurface agrees locally with the surface of simultaneity of the receiver today, and it is also, locally, a surface of simultaneity for any observer moving today with the "cosmological fluid."
The distance between emitter and observer today [the distance along the spatial geodesic of constant (t,theta,phi)(t, \theta, \phi) connecting ( t_(r),0,theta_(e),phi_(e)t_{r}, 0, \theta_{e}, \phi_{e} ) and ( {:t_(r),chi_(e),theta_(e),phi_(e))\left.t_{r}, \chi_{e}, \theta_{e}, \phi_{e}\right) ] can be read directly from the line element (29.4):
Using expression (29.9) for chi_(e)\chi_{e}, one finds
{:(29.12')ℓ=a(t_(r))int_(t_(e))^(t_(r))a^(-1)dt:}\begin{equation*}
\ell=a\left(t_{r}\right) \int_{t_{e}}^{t_{r}} a^{-1} d t \tag{29.12'}
\end{equation*}
where definitions (29.1) for the Hubble constant H_(o)H_{o} and the deceleration parameter q_(o)q_{o} have been used. Putting this expression into equation (29.12') and integrating, one finds for the distance the expression
This is the "distance-redshift relation" for the standard big-bang model of the universe.
By comparing this distance-redshift relation with astronomical observations (see Box 29.4, which is bes d after the next section), Allan Sandage (1972a) obtains a Hubble constant of
{:(29.18)H_(o)^(-1)=(18+-2)xx10^(9)" years. ":}\begin{equation*}
H_{o}^{-1}=(18 \pm 2) \times 10^{9} \text { years. } \tag{29.18}
\end{equation*}
(Note: 1Mpc-=1 \mathrm{Mpc} \equiv one Megaparsec is 3.26 xx10^(6)3.26 \times 10^{6} light years, or 3.08 xx10^(24)cm3.08 \times 10^{24} \mathrm{~cm}.) The uncertainty of +-7kmsec^(-1)Mpc^(-1)\pm 7 \mathrm{~km} \mathrm{sec}^{-1} \mathrm{Mpc}^{-1} quoted here is the "one-sigma" statistical uncertainty associated with the distance-redshift data. Systematic errors, not now understood, might be somewhat larger; but the true value of H_(o)H_{o} almost certainly is within a factor 2 of Sandage's value, 55kmsec^(-1)Mpc^(-1)55 \mathrm{~km} \mathrm{sec}^{-1} \mathrm{Mpc}^{-1}.
Result for distance-redshift relation
Measurement of Hubble constant H_(o)H_{o}
Note that, if Lambda=0\Lambda=0, then the "critical density" marking the dividing line between a "closed" universe and an "open" universe-i.e., between eventual recontraction and expansion forever-is
(As described in Box 29.1, rho > rho_("crit ")Longleftrightarrow\rho>\rho_{\text {crit }} \Longleftrightarrow "closed" Longleftrightarrow\Longleftrightarrow recontraction; rho < rho_("crit ")Longleftrightarrow\rho<\rho_{\text {crit }} \Longleftrightarrow "open" Longleftrightarrow\Longleftrightarrow expansion forever.) Comparison with the actual density will be delayed until $29.6.
The distance measurements are not accurate enough to yield useful information about the deceleration parameter, q_(0)q_{0}.
§29.4. THE MAGNITUDE-REDSHIFT RELATION; MEASUREMENT OF THE DECELERATION PARAMETER
Information about q_(o)q_{o} is best obtained by comparing the apparent magnitudes of galaxies with their redshifts.
In astronomy one defines the apparent (bolometric) magnitude, mm, of an object by the formula
where SS is the flux of energy (energy per unit time per unit area) that arrives at Earth from the object. [Of course, one cannot measure the flux over the entire wavelength range 0 < lambda < oo0<\lambda<\infty; so one distinguishes various apparent magnitudes ( m_(U),m_(B),m_(V),dotsm_{U}, m_{B}, m_{V}, \ldots ) corresponding to fluxes in various wavelength ranges (" U " -=\equiv "ultraviolet"; "B" 三"blue"; "V" 三"visual"). However, these subtleties are too far from gravitation physics to be treated here.]
Calculate the apparent magnitude for a galaxy of intrinsic luminosity LL and redshift zz. To simplify the calculation, put the emitter at the origin of the space coordinates ( chi_(e)=0\chi_{e}=0 ); and put the Earth at ( chi_(r),theta_(r),phi_(r)\chi_{r}, \theta_{r}, \phi_{r} ). (Note the reversal of locations compared to redshift calculation of §29.2\S 29.2.) On Earth, place a photographic plate of area AA perpendicular to the incoming light. Then at time t_(r)t_{r} the plate is a tiny segment of a spherical two-dimensional surface ( t=t_(r),chi=chi_(r);thetat=t_{r}, \chi=\chi_{r} ; \theta and phi\phi vary) about the emitting galaxy. The total area of the 2 -sphere surrounding the galaxy is
The plate catches a fraction A//AA / A of the energy that pours out through the 2 -sphere.
If there were no redshift, the power crossing the entire 2 -sphere at time t_(r)t_{r} would be precisely the luminosity of the emitter at time t_(e)t_{e}. However, the redshift modifies this result in two ways. (1) The energy of each photon that crosses the 2 -sphere is smaller, as measured in the local Lorentz frame of the fluid there, than it was as measured by the emitter:
(2) Two photons with the same theta\theta and phi\phi, which are separated by a time Deltat_(r)\Delta t_{r} as measured by an observer stationary with respect to the "cosmological fluid" at the 2 -sphere, were separated by a shorter time Deltat_(e)\Delta t_{e} as measured by the emitter:
The luminosity, LL, as measured at the source, is the sum of the energies E_("em. "J)E_{\text {em. } J} of the individual photons (labeled with the index JJ ) emitted in a time interval Deltat_(e)\Delta t_{e}, divided by Deltat_(e)\Delta t_{e} :
where RR is the "radius of curvature" of the 2 -sphere surrounding the emitter and passing through the receiver at the time of reception,
{:(29.28)R-=a_(o)Sigma(chi_(r)-chi_(e))={[a_(o)sinh(chi_(r)-chi_(e))," if "k=-1","],[a_(o)[chi_(r)-chi_(e)]," if "k=0","],[a_(o)sin(chi_(r)-chi_(e))," if "k=+1]:}:}R \equiv a_{o} \Sigma\left(\chi_{r}-\chi_{e}\right)= \begin{cases}a_{o} \sinh \left(\chi_{r}-\chi_{e}\right) & \text { if } k=-1, \tag{29.28}\\ a_{o}\left[\chi_{r}-\chi_{e}\right] & \text { if } k=0, \\ a_{o} \sin \left(\chi_{r}-\chi_{e}\right) & \text { if } k=+1\end{cases}
[recall: chi_(e)\chi_{e} is 0 according to the present conventions, and a_(o)=a(t_(r))a_{o}=a\left(t_{r}\right) ]. The corresponding apparent magnitude [equation (29.20)] is
In order to relate the apparent magnitude to the redshift of the emitter, one must express the quantity RR in terms of zz. From equation (29.7) for the photon propagation (with sign reversed because positions of receiver and emitter have been reversed), one knows that
Equations (4) to (6) of Box 29.1, and (27.40), determine the function dt//dad t / d a in terms of a//a(t_(r))a / a\left(t_{r}\right) and the constants H_(o),q_(o),sigma_(0)H_{o}, q_{o}, \sigma_{0}. By inserting that result into equation (29.31) and integrating, one obtains chi_(r)-chi_(e)\chi_{r}-\chi_{e} in terms of the redshift zz and the cosmological parameters H_(o),q_(o),sigma_(o)H_{o}, q_{o}, \sigma_{o} : chi_(r)-chi_(e)=|1+q_(o)-3sigma_(o)|^(1//2)int_(1)^(1+z)(du)/([2sigma_(o)u^(3)+(1+q_(o)-3sigma_(o))u^(2)+sigma_(o)-q_(o)]^(1//2))\chi_{r}-\chi_{e}=\left|1+q_{o}-3 \sigma_{o}\right|^{1 / 2} \int_{1}^{1+z} \frac{d u}{\left[2 \sigma_{o} u^{3}+\left(1+q_{o}-3 \sigma_{o}\right) u^{2}+\sigma_{o}-q_{o}\right]^{1 / 2}}.
The 2 -sphere radius of curvature RR is obtained by inserting this expression into the equation
[equation (29.28), with a_(o)a_{o} evaluated by equation (5) of Box 29.1].
Equations (29.29) and (29.32) determine the apparent magnitude, mm, in terms of redshift, zz.
For the case of vanishing cosmological constant ( sigma_(o)=q_(o);Lambda=0\sigma_{o}=q_{o} ; \Lambda=0 ), the integral (29.32a) can be expressed in terms of elementary functions, yielding
(Note: the factor 1.086 is actually 2.5//ln 102.5 / \ln 10.) A power-series solution for nonzero Lambda\Lambda (for sigma_(o)!=q_(o)\sigma_{o} \neq q_{o} ) reveals a dependence on sigma_(o)\sigma_{o} only at O(z^(2))O\left(z^{2}\right) and higher:
{:(29.35a)R~~H_(o)^(-1)z[1-(1)/(2)(1+q_(o))z+(" corrections of "O(z^(2))" depending on "sigma_(o)" and "q_(o))]:}\begin{equation*}
R \approx H_{o}^{-1} z\left[1-\frac{1}{2}\left(1+q_{o}\right) z+\left(\text { corrections of } O\left(z^{2}\right) \text { depending on } \sigma_{o} \text { and } q_{o}\right)\right] \tag{29.35a}
\end{equation*}
Sheldon (1971) gives the exact solution for Lambda!=0\Lambda \neq 0 in terms of the Weierstrass elliptic function. Refsdal et al. (1967) tabulate and plot the exact solution.
By comparing the theoretical magnitude-redshift relation (29.35b) with observations of the brightest galaxies in 82 clusters, Allan Sandage (1972a,c,d) obtains the
Measurement of deceleration parameter, q_(o)q_{o} following value for the deceleration parameter:
{:(29.36)q_(o)=1.0+-0.5","quad" if "sigma_(o)=q_(o)(" i.e. "Lambda=0):}\begin{equation*}
q_{o}=1.0 \pm 0.5, \quad \text { if } \sigma_{o}=q_{o}(\text { i.e. } \Lambda=0) \tag{29.36}
\end{equation*}
(Note: 0.5 is the "one-sigma" uncertainty. Sandage estimates with 68 per cent confidence that 0.5 < q_(o) < 1.50.5<q_{o}<1.5, and with 95 per cent confidence that 0 < q_(o) < 20<q_{o}<2 providing unknown evolutionary effects are negligible.) The observations leading to this result and the uncertainties due to evolutionary effects are described in Box 29.4. Box 29.5 gives a glimpse of Edwin Hubble, the man who laid the foundations for such cosmological measurements.
(continued on page 794)
Box 29.4 MEASUREMENT OF HUBBLE CONSTANT AND DECELERATION PARAMETER
I. Hubble Constant, H_(o)H_{o}
A. Objective: To measure the constant H_(o)H_{o} by comparing observational data with the distance-redshift relation
Here ℓ\ell is distance from Earth to source today; and zz is redshift of source as measured at Earth.
B. Key Difficulty: This distance-redshift relation does not apply to stars in our Galaxy: the Galaxy is gravitationally bound and therefore is impervious to the universal expansion. Nor does the distance-redshift relation apply to the separations between our Galaxy and nearby galaxies (the "local group"); gravitational attraction between our Galaxy and its neighbors is so great it perturbs their motions substantially away from universal expansion. Only on
Box 29.4 (continued)
scales large enough to include many galaxies (scales where each galaxy or cluster of galaxies can be thought of as a "grain of dust," with the grains distributed roughly homogeneously)-only on such large scales should the distance-redshift relation hold with good accuracy. But it is very difficult to obtain reliable measurements of the distances ℓ\ell to galaxies that are so far away!
C. Procedure by which H_(o)H_{o} has been measured [Sandage and Tamman, as summarized in Sandage (1972a)]:
Cepheid variables are pulsating stars with pulsation periods (as measured by oscillations in light output) that are very closely correlated with their luminosities LL-or, equivalently, with their absolute (bolometric) magnitudes, MM :
{:[(1)M-=((" apparent magnitude star would have were it at a ")/(" distance of "10" parsecs "=32.6" light years "))],[=-2.5log_(10)(L//3.0 xx10^(35)ergsec^(-1))]:}\begin{align*}
M & \equiv\binom{\text { apparent magnitude star would have were it at a }}{\text { distance of } 10 \text { parsecs }=32.6 \text { light years }} \tag{1}\\
& =-2.5 \log _{10}\left(L / 3.0 \times 10^{35} \mathrm{erg} \mathrm{sec}^{-1}\right)
\end{align*}
[see equation (29.20).] By measurements within our Galaxy, astronomers have obtained the "period-luminosity relation" for cepheid variables.
2. Cepheid variables are clearly visible in galaxies as far away as ∼4Mpc\sim 4 \mathrm{Mpc} (4 Megaparsecs -=4xx10^(6)\equiv 4 \times 10^{6} parsecs). In each such galaxy one measures the periods of the cepheids; one then infers their absolute magnitudes MM from the period-luminosity relation; one measures their apparent magnitudes mm; and one then calculates their distances ℓ\ell from Earth using the relation
By this means one obtains the distances ℓ\ell to all galaxies within ∼4Mpc\sim 4 \mathrm{Mpc} of our own. Unfortunately, such galaxies are too close to participate cleanly in the universal expansion. (They include only the "local group," the "M81 group," and the "south polar group.") Thus, one must push the distance scale out still farther before attempting to measure H_(0)H_{0}.
3. Galaxies of types Sc,Sd,Sm\mathrm{Sc}, \mathrm{Sd}, \mathrm{Sm}, and Ir within ∼4Mpc\sim 4 \mathrm{Mpc} contain huge clouds of ionized hydrogen, which shine brightly in "H alpha\alpha light." These clouds, called "H II regions," exhibit a very tight correlation between diameter DD of the H II region and luminosity LL of the galaxy (or, equivalently, between DD and absolute magnitude of galaxy, MM ). In fact, for a given galaxy luminosity LL, the fractional spread in H II diameters is sigma(Delta D//D)\sigma(\Delta D / D)~~0.12\approx 0.12. Using (a) the distances ( ≲4Mpc\lesssim 4 \mathrm{Mpc} ) to these galaxies as determined via cepheid variables, (b) the apparent magnitudes of the galaxies, and (c) the angular diameters of H II regions in the galaxies, one calculates
the actual H II diameters DD and galaxy luminosities LL, and thereby obtains the "diameter-luminosity relation" D(L)D(L).
4. H II regions are large enough to be seen clearly in galaxies as far away as ∼60Mpc\sim 60 \mathrm{Mpc}. By measuring the H II angular diameters alpha=D//ℓ\alpha=D / \ell and galaxy apparent (bolometric) magnitudes
and by combining with the diameter-luminosity relation, one obtains the distances ℓ\ell to all galaxies of types Sc,Sd,Sm\mathrm{Sc}, \mathrm{Sd}, \mathrm{Sm}, and Ir which possess H II regions and lie within ∼60Mpc\sim 60 \mathrm{Mpc} of Earth. Unfortunately, this is still not far enough away for local motions to be negligible compared with the universal expansion.
5. Within ∼60Mpc\sim 60 \mathrm{Mpc} reside enough galaxies of type Sc I for one to discover that their luminosities (absolute magnitudes) are rather constant (difference in LL from one Sc I galaxy to another ≲50\lesssim 50 per cent). Using the distances to such Sc I galaxies, as measured via H II regions, and using measurements of their apparent magnitudes, one calculates their universal absolute magnitude (measured photographically) to be M_(pg)=-21.2M_{p g}=-21.2.
The Sc I galaxy M101 at a distance ℓ∼3Mpc\ell \sim 3 \mathrm{Mpc} from Earth, as photographed with the 200 -inch telescope. (Courtesy of Hale Observatories)
6. One then examines all known Sc I galaxies with distances greater than ∼70Mpc\sim 70 \mathrm{Mpc}. For each, one measures the apparent magnitude and compares it with the universal absolute magnitude to obtain the distance ℓ\ell from Earth. And for each, one measures the redshift z=Delta lambda//lambdaz=\Delta \lambda / \lambda of the spectral lines. From the resulting redshift-distance relation-and taking into account the statistical uncertainties in all steps leading up to it-Sandage and Tamman (work carried out in 1965-1972) obtain the value H_(o)=dz//dℓ=55+-7(km//sec)Mpc^(-1)=1//[(18+-2)xx10^(9):}H_{o}=d z / d \ell=55 \pm 7(\mathrm{~km} / \mathrm{sec}) \mathrm{Mpc}^{-1}=1 /\left[(18 \pm 2) \times 10^{9}\right. years]. [For a review see Sandage (1972a).] The quoted error is purely statistical. Systematic errors are surely larger-but they almost surely do not exceed a factor 2 [i.e., 30 < H_(o) < 110(km//sec)Mpc^(-1)30<H_{o}<110(\mathrm{~km} / \mathrm{sec}) \mathrm{Mpc}^{-1} ].
Magnitude-redshift relation for Sc I galaxies at distances >= 70Mpc\geq 70 \mathrm{Mpc}. Solid line is a least-squares fit to the data; dotted line has the theoretical slope of 5. [From Sandage and Tamman.]
II. Deceleration Parameter, q_(0)q_{0}.
A. Objective: To measure the constant q_(o)q_{o} by comparing observational data with the magnitude-redshift relation:
[Note: This relation is valid even if the cosmological constant is nonzero, i.e., even if sigma_(o)!=q_(o)\sigma_{o} \neq q_{o}. Dependence on sigma_(o)\sigma_{o} occurs only at O(z^(2))O\left(z^{2}\right) and higher.]
B. Key Difficulty: One must use data for objects with the same absolute luminosity LL ("standard candles"). But one cannot measure LL at distances great enough for the effects of q_(o)q_{o} to show up.
C. The Search for a Standard Candle: One obvious choice for the standard candle would be Sc I galaxies, since they were found to all have nearly the same LL (see above). But they are not bright enough to be seen at distances great enough for effects of q_(o)q_{o} to show up. An alternative choice, quasars, are bright enough to be seen at very large redshifts ( zz as large as ∼3\sim 3 ). But their absolute luminosities have enormous scatter-or so one infers from the failure of quasars to fall on a straight line, even at small zz, in the magnitude-redshift diagram. The best choice is the brightest type of object that has small scatter in LL. Sandage (1972a,b,c) chooses the brightest galaxy in "recognized regular clusters of galaxies." Such clusters are composed predominately of E-type galaxies, and the brightest members are remarkably similar from one cluster
The E-type galaxy M87 at a distance ℓ∼11Mpc\ell \sim 11 \mathrm{Mpc} from Earth, as photographed with the 200 -inch telescope. (Courtesy of Hale Observatories)
to another (scatter in LL is ∼25\sim 25 per cent). The similarity shows up in their spectra and in the very precise straight lines they give when one plots angular diameter versus redshift (next page), or apparent magnitude versus redshift (next page), or angular diameter versus apparent magnitude.
Box 29.4 (continued)
Angular diameter versus redshift for brightest galaxy in recognized regular clusters. From Sandage (1972a,b). [These data are not sufficiently precise to yield useful information about q_(o)q_{o} and sigma_(a)\sigma_{a}; but improvements in 1973 may bring the needed precision; see $29.5.]
Magnitude versus redshift for brightest galaxy in recognized regular clusters. V_(26)-K_(V)-A_(v)\mathrm{V}_{26}-\mathrm{K}_{\mathrm{V}}-\mathrm{A}_{\mathrm{v}} is the apparent magnitude with certain corrections taken into account. The line plotted corresponds to sigma_(o)=q_(o)=1\sigma_{o}=q_{o}=1 (straight line of slope 5). From Sandage (1972a).
D. Procedure by which q_(o)q_{o} has been measured [Sandage (1972a,c)]:
Data on magnitude versus redshift have been gathered for the brightest galaxy in 82 recognized regular clusters (see above).
The data, when fitted with a straight line, show a slope of
{:(5)dm//dlog_(10)z=5.150+-0.268(rms):}\begin{equation*}
d m / d \log _{10} z=5.150 \pm 0.268(\mathrm{rms}) \tag{5}
\end{equation*}
by comparison with a theoretical slope of 5 .
3. The data, when fitted to the theoretical relation
The data are inadequate to determine sigma_(o)\sigma_{o} and q_(0)q_{0} simultaneously. [The O(z^(2))O\left(z^{2}\right) terms, which depend on sigma_(o)\sigma_{o}, play a significant role in the fit to the data. For a graphical depiction of their theoretical effects see Figure 2 of Refsdal et. al (1967).]
E. Evolutionary uncertainties
Sandage's fit of data to theory assumes that the luminosities of his "standard candles" are constant in time. If, because of evolution of old stars and formation of new ones, his galaxies were to dim by 0.09 magnitudes per 10^(9)10^{9} years, then galaxies 10^(9)10^{9} light-years away, which one sees as they were 10^(9)10^{9} years ago, would be 0.09 magnitudes brighter intrinsically than identical nearby galaxies. Correction for this effect would lower the most probable value of q_(0)q_{0} from 1 to 0 [Sandage (1972c)].
Knowledge of the evolution of galaxies in 1972 is too rudimentary to confirm or rule out such an effect. [See references cited by Sandage (1972c).]
Box 29.5 EDWIN POWELL HUBBLE
November 20, 1889, Marshfield MissouriSeptember 28, 1953, Pasadena, California
Edwin Hubble, at age 24, earned a law degree from Oxford University and began practicing law in Louisville, Kentucky. After a year of practice he became fed up and, in his own words, "chucked the law for astronomy, and I knew that even if I were second-rate or third-rate it was astronomy that mattered." He chose the University of Chicago and Yerkes Observatory as the site for his
astronomy education, and three years later (1917) completed a Ph.D. thesis on "Photographic Investigations of Faint Nebulae."
When Hubble entered astronomy, it was suspected that some nebulae lie outside the Galaxy, but the evidence was exceedingly weak. During the subsequent two decades, Hubble, more than anyone else, was responsible for opening to man's purview the extragalactic universe. Working with the 60 -inch and 100 -inch telescopes at Mount Wilson, Hubble developed irrefutable evidence of the extragalactic nature of spiral nebulae, elliptical nebulae, and irregular nebulae (now called galaxies). He devised the classification scheme for galaxies which is still in use today. He systematized the entire subject of extragalactic research: determining distance scales, luminosities, star densities, and the peculiar motion of our Galaxy; and obtaining extensive evidence that the laws of physics outside the Galaxy are the same as near Earth (in Hubble's words: "verifying the principle of the uniformity of nature"). He discovered and quantified the large-scale homogeneity of the universe. And-his greatest triumph of all!-he discovered the expansion of the universe.
The details of Hubble's pioneering work are best sketched in his own words:
"Extremely little is known of the nature of nebulae; and no classification has yet been suggested. .. The agreement [between the velocity of escape from a spiral nebula and that from our galaxy] is such as to lend some color to the hypothesis that the spirals are stellar systems at distances to be measured often in millions of light years."
(1920; Ph.d. THESIS; PUBLICATION DELAYED 3 YEARS BY WORLD WAR I)
"The present investigation [using Cepheid variables for the first time as an indicator of distances beyond the Magellanic clouds] identifies NGC 6822 as an isolated system of stars and nebulae of the same type as the Magellanic clouds, although somewhat smaller and much more distant. A consistent structure is thus reared on the foundation of the Cepheid criterion, in which the dimensions, luminosities, and densities, both of the system [NGC 6822] as a whole and its separate members, are of orders of magnitude which are thoroughly familiar. The distance is the only quantity of a new order. The principle of the uniformity of nature thus seems to rule undisturbed in this remote region of space."
(1925)
"Critical tests made with the 100 -inch reflector, the highest resolving power available, show no difference between the photographic images of the so-called condensations in Messier 33 and the images of ordinary galactic stars. . . . The
period-luminosity relation is conspicuous among the thirty-five Cepheids and indicates a distance about 8.1 times that of the Small Magellanic Cloud. Using
Shapley's value for the latter, the distance of the spiral is about 263,000 parsecs."
(1926a)
"[To the present paper (1926b)] is prefaced a general classification of nebulae the various types [of extragalactic nebulae] are homogeneously distributed over the sky. . . The data are now available for deriving a value for the order of the density of space. This is accomplished by means of the formulae for the numbers of nebulae to a given limiting magnitude and for the distance in terms of the magnitude. [The result is]
rho=1.5 xx10^(-31)" grams per cubic centimeter. "\rho=1.5 \times 10^{-31} \text { grams per cubic centimeter. }
This must be considered as a lower limit, for loose material scattered between the systems is entirely ignored. The mean density of space can be used to determine the dimensions of the finite but boundless universe of general relativity . . .
"The data . . . indicate a linear correlation between distances and velocities [for extragalactic nebulae]. Two solutions have been made, one using the 24 nebulae individually, the other combining them into 9 groups according the proximity in direction and distance. The results are . . . 24 objects: K=465+-50km//\mathrm{K}=465 \pm 50 \mathrm{~km} / sec per 10^(6)10^{6} parsecs; 9 groups: K=513+-60km//\mathrm{K}=513 \pm 60 \mathrm{~km} / sec per 10^(6)10^{6} parsecs. . . The outstanding feature, however, is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space."
(1929)*
Box 29.5 (continued)
"The velocity-distance relation is re-examined with the aid of 40 new velocities. . . The new data extend out to about eighteen times the distance available in the first formulation of the velocity-distance relation, but the form of the relation remains unchanged except for [Shapley's 10 per cent] revision of the unit of distance."
(1931), WITH M. L. HUMASON
Abstract
"Many ways of producing such effects [redshifts in extragalactic nebulae] are known, but of them all, only one will produce large redshifts without introducing other effects which should be conspicuous but actually are not found. This one known permissible explanation interprets redshifts as due to actual motion away from the observer."
(1934a)
"We now have a hasty sketch of some of the general features of the observable region as a unit. The next step will be to follow the reconnaissance with a survey - to repeat carefully the explorations with an eye to accuracy and completeness. The program, with its emphasis on methods, will be a tedious series of successive approximations."
(1934b)
Most of the remainder of Hubble's career was dedicated to this "tedious series of successive approximations." Shortly before Hubble's death the 200 -inch telescope went into operation at Palomar
Mountain; and Hubble's student, Alan Sandage, began using it in a continuation of Hubble's quest into the true nature of the universe. (See Box 29.4).
EXERCISES
Exercise 29.4. m(z)m(z) DERIVED USING STATISTICAL PHYSICS
Derive the magnitude-redshift relation using a statistical description of the photon distribution [cf. eq. (22.49) and associated discussion].
Exercise 29.5. DOPPLER SHIFT VERSUS COSMOLOGICAL REDSHIFT
(a) Consider, in flat spacetime, a galaxy moving away from the Earth with velocity vv, and emitting light that is received at Earth. Let the distance between Earth and galaxy, as measured in the Earth's Lorentz frame at some specific moment of emission, be rr; and let the Doppler shift of the radiation when it is eventually received be z=Delta lambda//lambdaz=\Delta \lambda / \lambda. Show that the flux of energy SS received at the Earth is related to the galaxy's intrinsic luminosity LL by
[Track-2 readers will find it most convenient to use the statistical formalism of equation (22.49).]
(b) Compare this formula for the flux with formula (29.27), where the redshift is of cosmological origin. Why is the number of factors of 1+z1+z different for the two formulas? [Mathematical answer: equation (6.28a) of Ellis (1971).]
§29.5. SEARCH FOR "LENS EFFECT' OF THE UNIVERSE
Curved space should act as a lens of great focal length. The curving of light rays has little effect on the apparent size of nearby objects. However, distant galaxiesgalaxies from a quarter of the way up to halfway around the universe-are expected to have greatly magnified angular diameters [Klauder, Wakano, Wheeler, and Willey (1959)]. To see a normal galaxy at such a distance by means of an optical telescope seems out of the question. However, radio telescopes resolve features in quasistellar sources and other radiogalaxies at redshifts of z=2z=2 or more. Moreover, paired radio telescopes at intercontinental distances (for example, Goldstone, California, and Woomera, Australia) resolve distant sources to better than 0^('').0010^{\prime \prime} .001 or 4.8 xx10^(-9)4.8 \times 10^{-9} radians or 15 lightyears for an object at a distance of 3xx10^(9)3 \times 10^{9} lightyears (Euclidean geometry temporarily being assumed). A radio telescope in space paired with a radio telescope on earth will be able to do even better on angular resolution. Will one be able to find any fiducial distance characteristic of any one class of objects that will serve as a natural standard of length, for very great distances ( z=2z=2 to z=3z=3 ) as well as for galaxies closer at hand? Perhaps not. However, it would seem unwise to discount this possibility, with all the advantages it would bring, in view of the demonstrated ability of skilled observers to find regularities elsewhere where one had no right to expect them in advance.
Let LL denote the actual length of a fiducial element (if any be found) in a galaxy; and let delta theta\delta \theta (radians!) denote the apparent length of the object, idealized as perpendicular to the line of sight, as seen by the observer. The ratio of these two quantities defines the "angle effective distance" of the source,
In flat space and for objects with zero relative velocity, this distance is to be identified with the actual distance, rr, to the source or with the actual time of flight, tt, of light from source to observer. The situation is changed in an expanding universe.
To calculate the angle effective distance as a function of redshift, place the Earth (receiver) at chi_(r)=0\chi_{r}=0; and place the object under study (emitter) at chi_(e)\chi_{e}. Let the fiducial length LL lie on the sphere at chi_(e)\chi_{e} (perpendicular to line of sight), and let it run from theta_(e)\theta_{e} to theta_(e)+delta theta\theta_{e}+\delta \theta [one end of fiducial element at (chi_(e),theta_(e),phi_(e))\left(\chi_{e}, \theta_{e}, \phi_{e}\right); other at (chi_(e),theta_(e)+delta theta,phi_(e))\left(\chi_{e}, \theta_{e}+\delta \theta, \phi_{e}\right) ]. Then
Here RR is given as a function of redshift of source, zz, and cosmological parameters H_(o),q_(o),sigma_(o)H_{o}, q_{o}, \sigma_{o}, by equations (29.32) in general, or by (29.33) if Lambda=0\Lambda=0. [Equation (29.38b) is modified if the beam preferentially traverses regions of low mass density ("vacuum between the galaxies"); see equation (22.37) and Gunn (1967).]
The hope for a fiducial length in distant objects
Angle effective distance defined
Angle effective distance as function of redshift
Figure 29.2.
Angle effective distance versus redshift for two typical cosmological models-one open (0 < sigma_(o)=q_(o)≪1)\left(0<\sigma_{o}=q_{o} \ll 1\right); the other closed ( sigma_(o)=q_(o)=1\sigma_{o}=q_{o}=1 ); both with zero cosmological constant; both with H_(o)^(-1)=18 xx10^(9)lyrH_{o}{ }^{-1}=18 \times 10^{9} \mathrm{lyr}.
Angle effective distance as a tool for determining whether universe is closed
Figure 29.2 shows angle effective distance as a function of redshift for a few selected choices of the relevant parameters. It is evident that the angle effective distance has a maximum for a redshift roughly of the order z∼1z \sim 1, provided that the universe is closed. However, there is a big difference if the universe is open (Figure 29.2). The rapid improvements taking place in radio astronomy make increasingly attractive the possibility it provides for testing whether the universe is closed, as Einstein argued it should be [Einstein (1950), pp. 107-108]. Moreover, even with optical telescopes, in 1973 one may be on the verge of measuring q_(o)q_{o} by studies of angle effective distance: preliminary studies [Sandage (1972b)] suggest that the optical size of the brightest E-type galaxies may be a usable fiducial length.
§29.6. DENSITY OF THE UNIVERSE TODAY
It is exceedingly difficult to measure the mean density rho_(mo)\rho_{m o} of the universe today. A large amount of matter may be in forms that astronomers have not yet managed to observe (intergalactic matter, black holes, etc.). Therefore, the best one can do is to add up all the luminous matter in galaxies and regard the resulting number as a lower limit on rho_(mo)\rho_{m o}. Even adding up the luminous matter is a difficult and risky task, so difficult that even today no analysis is more definitive than the classic work of Oort (1958). [See, however, the very detailed review of the problem in Chapter 4 of Peebles (1971)]. Oort's result is
{:(29.40)sigma_(o) >= 0.02quad" (independent of the value of "H_(o)" ). ":}\begin{equation*}
\sigma_{o} \geq 0.02 \quad \text { (independent of the value of } H_{o} \text { ). } \tag{29.40}
\end{equation*}
As an example (albeit an atypical one) of the danger inherent in any such estimate, Oort points to the Virgo cluster of galaxies. If the Virgo cluster is not gravitationally bound, then its ∼2,500\sim 2,500 galaxies will go flying apart, destroying any semblance of a cluster, in about one billion years. If it is gravitationally bound, then the mean velocity of its galaxies relative to each other, when combined with the virial theorem, yields an estimate of the cluster's total mass. That estimate is 25 times larger than the value one gets by Oort's method of adding up the luminous mass of the cluster.
Although one has no definitive evidence for or against large amounts of matter (enough to close the universe) in intergalactic space, one has tentative indirect limits: (1) If Lambda=0\Lambda=0 (in accord with Einstein), then sigma_(o)=q_(o)\sigma_{o}=q_{o}; so Sandage's value of q_(o) <=q_{o} \leqq 1 -stretched to q_(0) < 10q_{0}<10 under the most wild of assumptions about galaxy evolu-tion-implies
(2) Gott and Gunn (1971) point out that, if the density of gas in intergalactic space were >= 10^(-30)g//cm^(3)\geq 10^{-30} \mathrm{~g} / \mathrm{cm}^{3} (i.e., if sigma_(o)\sigma_{o} were >= 0.1\geq 0.1 ), one would expect gas falling into the Coma cluster of galaxies to form a shock wave, which would emit large amounts of X-rays. From the current X-ray observations, one can place a limit on the amount of such infalling matter-and therefrom a limit
on the density of gas in intergalactic space. But these limits, like others obtained in other ways [see Chapter 4 of Peebles (1971) for a review] are far from definitive; they depend too much on theoretical calculations to make one feel fully comfortable.
§29.7. SUMMARY OF PRESENT KNOWLEDGE ABOUT COSMOLOGICAL PARAMETERS
The best data available in 1973 [equations (29.18), (29.36), (29.40)] reveal
{:[H_(o)^(-1)=(18+-2)xx10^(9)" years "],[(29.41)q_(o)=1+-0.5(" one-sigma ")quad" if "sigma_(o)=q_(o)(Lambda=0)],[sigma_(o) >= 0.02]:}\begin{gather*}
H_{o}^{-1}=(18 \pm 2) \times 10^{9} \text { years } \\
q_{o}=1 \pm 0.5(\text { one-sigma }) \quad \text { if } \sigma_{o}=q_{o}(\Lambda=0) \tag{29.41}\\
\sigma_{o} \geq 0.02
\end{gather*}
for the observational parameters of the universe. These numbers are inadequate to reveal whether the universe is closed or open, and whether it will continue to expand forever or will eventually slow to a halt and recontract.
If one is disappointed in this lack of knowledge, one can at least be consoled by the following. (1) There is excellent agreement between theory and observation for the linear (lov-z) parts of the distance-redshift, magnitude-redshift, and angular diameter-redshift relations (Box 29.4). (2) There is remarkably good agreement between (a) the age of the universe ( 18 billion years if q_(o)=sigma_(o)≪1;12q_{o}=\sigma_{o} \ll 1 ; 12 billion years if q_(o)=sigma_(o)=(1)/(2)q_{o}=\sigma_{o}=\frac{1}{2} ) as calculated from the measured value of H_(o)H_{o}; (b) the ages of the
Summary of observational parameters of universe
Some quantitative triumphs of cosmology
The bright prospects for observational cosmology
oldest stars (∼10 xx10^(9):}\left(\sim 10 \times 10^{9}\right. years) as calculated by comparing the theory of stellar evolution with the properties of the observed stars; (c) the time ( ∼9\sim 9 billion years) since nucleosynthesis of the uranium, thorium, and plutonium atoms that one finds on Earth, as calculated from the measured relative abundances of various nucleides; and (d) the ages ( 4.6 billion years) of the oldest meteorites and oldest lunar rock samples, as calculated from measured relative abundances of other nucleides. For further detail see, e.g., Sandage (1968, 1970), Wasserburg et al. (1969), Wasserburg and Burnett (1968), and Fowler (1972). (3) Observations of the cosmic microwave radiation and measurements of helium abundance are now capable of giving direct information about physical processes in the universe at redshifts z≫1z \gg 1 (Chapter 28). (4) One may yet find "fiducial lengths" in radio sources, visible out to z >= 1z \geq 1, with which to measure q_(o)q_{o} and sigma_(o)\sigma_{o} by the angle-effective-distance method ( $29.5\$ 29.5 ). (5) The enigmas of the nature of quasars and of their peculiar distribution with redshift (great congregation at z∼2z \sim 2; absence at z >= 3z \geq 3 ) may yet be cracked and may yield, in the process, much new information about the origin of structure in the universe (Box 28.1). (6) The next decade may well bring as many great observational surprises, and corresponding new insights, as has the last decade.
EXERCISES
Exercise 29.6. SOURCE COUNTS
Suppose that one could find (which one cannot) a family of light or radio sources that (1) are all identical with intrinsic luminosities LL, (2) are distributed uniformly throughout the universe, and (3) are born at the same rate as they die so that the number in a unit comoving coordinate volume is forever fixed.
(a) Show that the number of such sources N(z)N(z) with redshifts less than zz, as observed from Earth today, would be
[Answer: See §15.7\S 15.7 of Robertson and Noonan (1968).]
Exercise 29.7. COSMIC-RAY DENSITY (Problem devised by Maarten Schmidt)
Suppose the universe has contained the same number of galaxies indefinitely into the past. Suppose further that the cosmic rays in the universe were created in galaxies and that a negligible fraction of them have been degraded or lost since formation. Derive an expression for the average density of energy in cosmic rays in the universe today in terms of: (1) the number density of galaxies, N_(o)N_{o}, today; and (2) the nonconstant rate, dE//dzd E / d z, at which the average galaxy created cosmic-ray energy during the past history of the universe. [At redshift zz in range dzd z, the average galaxy liberates energy (dE//dz)dz(d E / d z) d z into cosmic rays.]
Exercise 29.8. FRACTION OF SKY COVERED BY GALAXIES
Assume that the redshifts of quasars are cosmological. Let the number of galaxies per unit physical volume in the universe today be N_(o)N_{o}, and assume that no galaxies have been created or destroyed since a redshift of >= 7\geq 7. Let DD be the average angular diameter of a galaxy. Calculate the probability that the light from a quasar at redshift zz, has passed through at least one intervening galaxy during its travel to Earth. [For a detailed discussion of this problem, see Wagoner (1967).]
(continued on page 711)
*"Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space-bounded, universe:
"1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe.
"2. The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained, to a first approximation, in the equations of the theory of relativity; . . . But this idea of Mach's corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe" [Einstein (1950), pp. 107-108].
Many workers in cosmology are skeptical of Einstein's boundary condition of closure of the universe, and will remain so until astronomical observations confirm it.
^(**)H_(o){ }^{*} H_{o} is predicted to be independent of the choice of galaxy insofar as local motions are unimportant, and insofar as the difference between recession velocity now and recession velocity at the time when the light was emitted is unimportant. The latter condition is well fulfilled by galaxies close enough to admit of the necessary measurement of distance, for they have redshifts only of the order of z∼0.1z \sim 0.1 and less (little lapse of time between emission of light and its reception on earth; therefore little change in recession velocity between then and now; see $29.3\$ 29.3 and Box 29.4 for a fuller analysis).
"Isotropy of universe" defined
*With this choice of spatial coordinates, the spacetime metric reads
This is often called the "Robertson-Walker line element," because Robertson (1935,1936)(1935,1936) and Walker (1936) gave the first proofs that it describes the most general homogeneous and isotropic spacetime geometry.
This box is based largely on the biography of Hubble by Mayall (1970).
*Hubble's value of KK (the "Hubble constant") was later revised downward by the work of Baade and Sandage; see section titled The Hubble Time in Box 27.1.
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